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.