Why study Higher Sheaf Cohomology? The classical lore is that $H^1(X,\mathcal F)$ is obstruction to lifting local data to global data. However I don't understand why one would want to compute $H^3(X,\mathcal F), H^4(X,\mathcal F), \cdots$.

For complex manifold $X$, $H^1(X,\mathcal O),H^{0,1}(X)$ both represent obstruction to local-to-global lifting of holomorphic functions. This in particular allows one to determine whether Mittag-Leffler problem can be solved. $H^1(X,\mathcal O)=0$ implies local solutions can be modified to identify a global solution, and $H^{0,1}(X)=0$ implies that local solutions can be multiplied by a smooth bump function, after which $\bar\partial$-exactness kicks in to save the day. 
However, what does $H^q(X,\Omega^p)$ or $H^{p,q}(X)$ mean (which are the same by Dolbeault theorem)? $H^1(X,\Omega^p)$ or $H^{p,1}(X)$ should represent the same local-to-global problem for holomorphic $p$-forms, but what about, say, $H^3(X,\Omega^p)$?
Sure, one can talk about Cech cohomology for a good cover; $H^3(X,\mathcal F)$, for example, is about lifting sections defined on quadruple-intersections to triple-intersections. That's fine and all, but that doesn't sound very compelling to me. It seems to me that lifting sections on $q$-fold intersections to sections on $(q-1)$-fold intersections doesn't really solve any natural, interesting problems that arise independently of the formalisms introduced (for example, Lefschetz fixed point formula solves the problem of counting fixed points, and this is defined independently of singular homology and therefore I'd consider this a very compelling reason to study singular homology groups).
Similar situation appears when we study Chern class and line bundles. The Bockstein morphism for exponential sheaf sequence $H^1(X,\mathcal O^\times) \rightarrow H^2(X,\mathbb Z)$ is precisely taking Chern class for line bundle, so it helps to know things like $H^1(X,\mathcal O)=H^2(X,\mathcal O)=0$, which allow us to classify line bundles for manifolds like $\mathbb {CP}^n$. However, there does not seem to be a reason to care about $H^3(X,\mathcal O)$, etc.

Note: I have checked out a similar question. Here, one of the answers point out that for sheaf $\mathcal F$, if we can find an acyclic sheaf $\mathcal A$ such that $\mathcal F$ is a subsheaf of $\mathcal A$, then
$$0\rightarrow \mathcal F \rightarrow \mathcal A \rightarrow \mathcal A/\mathcal F \rightarrow 0 $$
is exact and therefore by long exact sequence coming from this,
$$ H^{p}(X,\mathcal F) \cong H^{p-1}(X,\mathcal A / \mathcal F)$$
and therefore higher cohomology groups can be understood as obstruction ($H^1$), and actually even global section ($H^0$) of $\mathcal A_1/(\mathcal A_2 / \cdots (\mathcal A_p /\mathcal F)\cdots )$. This just mystifies the issue further for me, somewhat, largely because I can't think of a canonical choice of such acyclic $\mathcal A$ and therefore I can't interpret the meaning of local-to-global lifting of iterated-quotient sheaves.
 A: In addition to the other great answers, there are some proofs in algebraic geometry where we establish a result about $H^q(X, \mathcal{E})$ by descending induction on $q$: The base case is $q = \dim X$ and then we show Case $q$ implies Case $q-1$. Even if you only care about the result for $H^0$, this may be the best route. 
The big example is Grothendieck's proof that proper pushforwards of coherent sheaves are coherent. See the discussion before Cor 18.1.15 in Vakil, and related examples throughout Chapter 18.
A: I think these kinds of lifting-based statements are the wrong place to start. For me these arguments about lifting and such are most convincing as answers to questions like "What are cohomology groups?". For instance $H^1(X,\mathcal O_X)$ is the tangent space to the moduli space of line bundles on $X$. But that don't really explain why to study it. How much do you really care about the Mittag-Leffler problem? Probably not enough to base your entire mathematical career around it. But a huge fraction of modern mathematicians have based their mathematical careers around studying cohomology or generalisations of it.
Instead, as you suggest, what motivates the study of cohomology are its immensely powerful applications. For Dolbeaut cohomology and coherent sheaf cohomology in particular, I would say the premier applications are to classification-type problems in algebraic geometry. 
First, Dolbeaut cohomology groups provide natural invariants (the Hodge numbers) that can be used to distinguish algebraic varieties.
Second, the machinery of sheaf cohomology is crucial in answering all sorts of geometric questions about line bundles and other geometric structures. A good example might be the proof of the  Hodge index theorem via Hirzebruch-Riemann-Roch, as in Hartshorne. We need all the cohomology groups to define the Euler characteristic, and the Euler characteristic is such a nice invariant that there is a simple formula for it, and this simple formula helps us understand purely geometric questions about intersection of curves.
I would say that at a typical algebraic geometry seminar the majority of the talks involve higher sheaf cohomology at some point, but most talks are not devoted to answering a question expressed in terms of cohomology, so one could find many examples there.
Third, there are the applications via Hodge theory. The natural isomorphism between the singular cohomology and the Dolbeaut cohomology itself has nontrivial structure which is related in a highly nontrivial way to the geometry and arithmetic of the variety. In particular it is supposed to tell you about algebraic cycles - this is the Hodge conjecture. While that is open, many interesting facts are known - e.g. the K3 surfaces are completely determined by their Hodge structure.
Studying how these Hodge structures vary in a family of varieties leads to all sorts of interesting theory, which again can be used to solve purely geometric questions - the statement I remember off the top of my head is that varieties of general type cannot have a nowhere vanishing one-form (Popa and Schnell).
And this is not even getting into the very interesting applications of other cohomology theories for algebraic varieties, such as etale cohomology.
A: Do you agree that something like $z_1^2z_2 dz_1 \wedge d\bar{z}_3 + z_3 dz_1 \wedge dz_2$ is a natural sort of thing to integrate over a complex 2 dimensional submanifold of $C^3$?  I think that these forms are "intrinsically interesting", just because we want to do integral calculus.
Closed forms are always locally exact, but may or may not be globally exact.  It seems intrinsically interesting to investigate those forms which are locally but not globally exact.
I guess I am just saying that these are the natural questions you would ask when you start doing integral calculus on complex spaces?
What sort of application are you looking for?
A: I think you're absolutely right that the function $(i\in \mathbb N)\mapsto $interestingness($H^i$) is a rapidly decreasing function. I heard that Gel$'$fand compared it to the successive derivatives of position: we care about speed and acceleration, possibly about jerk, not so much about jounce or beyond.
However, you very often have to have those $H^i$ if you want to define Euler characteristic (in whatever context), so even if you don't want to compute a particular $H^i$ you want it around theoretically.
In the context of coherent sheaves, very often the approach is to compute Euler characteristic, give some reason that higher sheaf cohomology vanishes, and conclude that one has computed $H^0$. Serre gave a talk (around 1998) in which he explained that "It was once believed that the only good sheaf cohomology is dead sheaf cohomology" but "nowadays this is not the [politically] correct view."
A: I can't answer the question strictly within the framework of algebraic geometry, but in representation theory of semisimple algebraic groups there is a partial answer: sheaf cohomology groups of line bundles on the flag variety $G/B$ afford natural representations of the given group (as in the theorems of Borel-Weil and Bott), and in prime characteristic the higher degree cohomology sometimes leads to mysterious new representations larger than Weyl modules but still having a unique highest weight with multiplicity 1.    It's still an open-ended problem, but the small rank case of $G_2$ illustrates it well: see for example a recent paper by Andersen-Kaneda.
Since 1977 Henning Andersen has found many interesting patterns in higher cohomology, which so far agree with my suspicion that these results are determined by Kazhdan-Lusztig theory (involving inverse Kazhdan-Lusztig polynomials) for affine Weyl groups.   See for example an old paper on cohomology of $G/B$ in characteristic $p$ here.
A: It is true that the primary object of interest is $H^0$—global sections. 
I will give you four reasons for the importance of higher cohomology: use of exact sequences, the Riemann-Roch problem, the Cousin problem, and the GAGA theorems. I conclude the answer with discussing the problem of representing geometrically cohomology classes.


*

*Exact sequences


As you observed, cohomology reveals useful when some exact sequences of sheaves do not lead to exact sequences at the level of global sections – something we now understand as non-vanishing of a first cohomology group. Some classical restriction theorems of Algebraic geometry can be seen as stating the vanishing of a first cohomology group. But once you accept exact sequences with 6 terms involving $H^0$ and $H^1$, why don't go further and extend it with $H^2$, etc.?


*The Riemann-Roch problem


Historical 19th century problems of algebraic geometry asked to construct holomorphic/meromorphic functions with prescribed zeroes/poles on the complex plane $\mathbf C$, on Riemann surfaces, or on $\mathbf C^n$, or on complex algebraic manifolds. That's the origin of the Riemann problem and the Riemann inequality : if $D$ is a divisor on a Riemann surface $X$ of genus $g$, meromorphic functions $f$ with poles at most $D$ are global sections of a sheaf $\mathscr O_X(D)$, and the Riemann inequality states that 
$$ \dim (\Gamma(\mathscr O_X(D)))\geq \deg(D)+1-g.$$
The Riemann inequality  has been made more precise thanks to Roch, leading to the Riemann-Roch equality: if $K$ is a canonical divisor on $X$, then 
$$ \dim (H^0(\mathscr O_X(D))) = \deg(D)+1-g + \dim (H^0(\mathscr O_X(K-D)).$$
The picture gets messier on surfaces where Italian geometers had proved an inequality of the form
$$ \dim(H^0(\mathscr O_X(D)))+\dim(H^0(\mathscr O_X(K-D))) \geq \frac12 D\cdot (K-D) + 1 + p_a, $$
wher $p_a$ is the arithmetic genus.
As Hartshorne writes (Algebraic geometry, p. 363) : The difference is called the superabundance, because, before the invention of cohomology, it was the defect of validity of the corresponding equality.
Having a natural equality is the content of the Riemann-Roch theorem for surfaces offers a more precise theorem involving Euler-Poincaré characteristics 
$$ \chi(\mathscr O_X(D)) = \dim(H^0(\mathscr O_X(D)))- \dim(H^1(\mathscr O_X(D))+\dim(H^2(\mathscr O_X(D))) = \frac12 D\cdot (K-D) + \chi(\mathscr O_X), $$
which implies the former formula thanks to the duality theorem
$$ H^2(\mathscr O_X(D)) \simeq H^0(\mathscr O_X(K-D))^\vee. $$
A similar interpretation holds in the case of Riemann surfaces, 
$$ \chi(\mathscr O_X(D))=\dim(H^0(\mathscr O_X(D)))-\dim(H^1(\mathscr O_X(D)) = \deg(D)+1-g, $$
and the duality theorem
$$ H^1(\mathscr O_X(D)) \simeq H^0(\mathscr O_X(K-D))^\vee. $$
The generalization in higher dimension has been proved by Hirzebruch and Grothendieck, but is still of the same form: a formula for the Euler-Poincaré characteristic $\chi(\mathscr O_X(D))$ in terms of intersection theory and characteristic classes. 
It has to be complemented with duality theorems (Serre, Grothendieck) and vanishing theorems (Serre, Kodaira), which allow to express the Euler-Poincaré characteristic in terms of more concrete objects, such as $H^0$.


*Cousin problems


The Cousin problem asks to construct meromorphic functions with prescribed poles on the complex plane. If one tries to generalize it,
one observes that its possibility lies in the vanishing of some cohomology groups, such as $H^1(X,\mathscr O_X^\times)$.


*The GAGA theorems


Here GAGA is not a pop star, but an acronym for Géométrie algébrique et géométrie analytique, the title of a famous paper by Serre. Some comparison theorems between those two geometric theories were well known: for example, any meromorphic function on the projective space $\mathbf P^n(\mathbf C)$ is a rational function, and any analytic closed subvariety  of $\mathbf P^n(\mathbf{C})$ can be defined by polynomials (Chow's theorem). Serre's generalization is sheaf theoretical: it says that the functor of analytification of algebraic coherent sheaves leads to an equivalence of categories between the categories of algebraic and analytic coherent sheaves. Its proof is cohomological and establishes that the analytification functor preserves cohomology groups, a fact that Serre proves by decreasing induction, starting with cohomology of high degree, where the result is trivial because both cohomology groups vanish in degrees strictly larger than the dimension. The most important statement, that of global sections — $H^0$ —, is obtained at the last step of the induction, hence is in some sense the most difficult.


*The quest for a geometric interpretation of cohomology


Maybe, a difficulty of cohomology lies in the lack of a geometrical interpretation. 
For $H^1$, this is quite easy to achieve: $H^1(X,\mathscr F)$ parameterizes principal homogeneous spaces under $\mathscr F$; as we saw above, this interpretation is already useful in the study of the Cousin problem. 
The description is much more complicated for $H^2$ — it is the topic of nonabelian cohomology, initiated by Grothendieck and Giraud.
I do not know however of similar geometric interpretations in higher degrees.
