Here is some of the general philosophy of birational invariants, at least those coming from (co)homology (I don't think this approach quite works for homotopical invariants.)
Philosophy. If $f \colon X \dashrightarrow Y$ is a birational map, consider the cycles $\bar \Gamma_f \subseteq X \times Y$ and $\bar \Gamma_{f^{-1}} \subseteq Y \times X$. Then $\bar\Gamma_{f^{-1}} \circ \bar \Gamma_f$ and $\bar\Gamma_f \circ \bar\Gamma_{f^{-1}}$ differ from the identity by a cycle supported on $D \times D$ and $E \times E$ respectively, where $D \subseteq X$ and $E \subseteq Y$ are divisors. (For a proof, see e.g. this answer.)
Since cycles "act on any cohomology theory", this has implications for the parts of cohomology that are acted upon trivially by cycles supported on $D \times D$. Let me give some examples.
Definition. The coniveau filtration on $H^i(X,\mathbb Z/n \mathbb Z)$ (singular or étale cohomology, assuming $(n,p) = 1$ if $k$ has positive characteristic) is defined by
$$N^r H^i(X,\mathbb Z/n\mathbb Z) = \left\{x\ \bigg|\ \begin{array}{cc} x|_{X-Y} = 0 \in H^i(X-Y,\mathbb Z/n\mathbb Z) \\ \text{ for some } Y \subseteq X \text{ with } \operatorname{codim}(Y,X) \geq r\end{array}\right\}.$$
Lemma. Let $Z \in \operatorname{CH}^n(X \times Y)$ by a cycle supported on $X \times E$. Then the image of the induced map
$$Z_* \colon H^*(X, \mathbb Z/n\mathbb Z) \to H^*(Y, \mathbb Z/n\mathbb Z)$$
lands in $N^1H^*(Y,\mathbb Z/n\mathbb Z)$.
Proof. If $x \in H^i(X,\mathbb Z/n\mathbb Z)$, then $Z_* x$ is supported on $E$. Thus, $(Z_* x)|_{Y-E} = 0$. $\square$
Examples. Let me sketch how this philosophy manifests itself in certain cohomology theories:
Hodge realisation. In the Hodge realisation, the coniveau $r$ part of the cohomology will land inside
$$H^{k-r,r}(X) \oplus \ldots \oplus H^{r,k-r}(X) \subseteq H^k(X,\mathbb C).$$
Indeed, an element of $H^k(X,\mathbb C)$ that vanishes on $X-Y$ with $\operatorname{codim}(Y,X) = r$ comes from the cohomology $H^{k-2r}(Y,\mathbb C)$ under the Gysin map, at least when $Y$ is smooth (in general, replace $Y$ by a resolution $\tilde Y \to Y$). This is a morphism of Hodge structures of bidegree $(r,r)$, which proves the claim.
This shows that $H^0(X,\Omega^k_X)$ and $H^k(X,\mathcal O_X)$ are birational invariants: $(\bar\Gamma_{f^{-1}} \circ \bar\Gamma_f) _* = \operatorname{Id} + Z$, with $Z$ supported on $D \times D$. By the lemma above, $Z_*$ maps everything into the coniveau $1$ part, so it acts as $0$ on $H^k(X,\mathcal O_X) \oplus H^0(X,\Omega^k_X)$. Hence, the composition
$$H^k(X,\mathcal O_X) \to H^k(Y,\mathcal O_Y) \to H^k(X,\mathcal O_X)$$
is the identity, and similarly for the opposite composition, as well as for $H^0(-,\Omega^k)$. $\square$
In this example, one can also use the explicit description of the Hodge diamond of a blow-up, and use weak factorisation. Or even use the elementary argument that $H^0(-,\Omega^k)$ is a birational invariant and invoke Hodge symmetry.
Frobenius realisation. Over a finite field $k$, we have the slope filtration on crystalline cohomology (after inverting $p$), which is the closest analogue to the Hodge decomposition on de Rham cohomology. This gives a filtration on
$$H^k_{\operatorname{crys}}(X/K) = H^k_{\operatorname{crys}}(X/W(k))[\tfrac{1}{p}]$$
whose successive subquotients are the spaces where Frobenius acts with slope $[i,i+1)$. These spaces can alternatively be described (cf. Illusie) as the de Rham-Witt cohomology
$$H^k_{\operatorname{crys}}(X/K)_{[i,i+1)} = H^i(X,W\Omega^{k-i}_X)[\tfrac{1}{p}],$$
which is why it is similar to the Hodge decomposition of de Rham cohomology in characteristic $0$.
It follows from Berthelot's work that the coniveau $r$ part has slopes $\geq r$ (see e.g. Lemma 2.1 of this paper by Hélène Esnault; the argument is identical to the one given above in the Hodge realisation: use purity of the Gysin maps). Thus, the slope $[0,1)$ part is a birational invariant (hence by slope symmetry, so is the slope $(k-1,k]$ part).
Explicitly, this says that the multiset of Frobenius-eigenvalues not divisible by $q = |k|$ is a birational invariant. This was actually proven in an elementary way by Ekedahl in 1983 (Sur le groupe fondamental d'une variété unirationelle).
Cohomology of $\mathcal O_X$ in positive characteristic. In this paper by Chatzistamatiou and Rülling, it is shown that $H^i(X,\mathcal O_X)$ is a birational invariant in positive characteristic. The main idea of the paper is to define a well-defined action of Chow groups on Hodge cohomology [with compact support] and use the philosophy above. (The main difficulty is to define pushforwards, which is where the supports come in.)
Chow group of zero-cycles. The group $\operatorname{CH}_0(X)$ is a birational invariant over any field. Indeed, it suffices to show that a cycle supported on $D \times D$ induces the zero map on $\operatorname{CH}_0(X)$. This is because we can move any zero-cycle to a cycle not meeting $D$. See this answer for more details.
Other examples. Low degree homology $H_2(X,\mathbb Z)$ "should only pick up things coming from $H_0(D, \mathbb Z)$" if we act by a cycle supported on $X \times D$. This is torsion-free, hence the torsion in $H_2$ is a birational invariant. Under universal coefficients, this gives the torsion in $H^3$.
Similarly, the $n$-torsion of the Brauer group is given by $H^2(X,\mathbb Z/n \mathbb Z)$ via the exact sequence
$$0 \to \mu_n \to \mathbb G_m \to \mathbb G_m \to 0.$$
Splitting this up into $n$-torsion in $H^3(-,\mathbb Z)$ and the part coming from $H^2(-,\mathbb Z)/n$, we see that it should be a birational invariant. (In positive characteristic, use $\mathbb Z_\ell$ instead of $\mathbb Z$, at least when $n = \ell$. There is also a characteristic-independent proof using Brauer-theoretic arguments.) (In characteristic $0$ we can bypass $\mathbb Z/n \mathbb Z$ by using the exponential sequence instead of the argument I gave here.)
