Reference request on birational invariance of Chow group of zero cycles of degree zero
A reference for birational equivalence of $CH_0$ is Fulton's Intersection Theory [1], Example 16.1.11. In the example, he makes the assumption that $k$ is algebraically closed, but he never uses it. Since the argument is fairly short, let me repeat it here.
Theorem. Let $k$ be a field, and let $X$ and $Y$ be smooth proper $k$-varieties. If $X$ and $Y$ are birational, then $CH_0(X) \cong CH_0(Y)$.
Proof. Let $f \colon X \dashrightarrow Y$ be a birational map. Let $\Gamma \subseteq X \times Y$ be the closure of the graph. Then $\Gamma$ defines maps \begin{align*} f_* \colon CH_0(X) &\to CH_0(Y) & f^* \colon CH_0(Y) &\to CH_0(X)\\ a &\mapsto \pi_{Y,*}(\Gamma \cdot \pi_X^* a), & b &\mapsto \pi_{X,*}(\Gamma \cdot \pi_Y^* b). \end{align*} The composition $f^* \circ f_*$ (resp. $f_* \circ f^*$) is given by the cycle $\Gamma^\top \circ \Gamma := \pi_{13, *} (\pi_{12}^* \Gamma \cdot \pi_{23}^* \Gamma^\top)$ on $X \times X$ (resp. by $\Gamma \circ \Gamma^\top$ on $Y \times Y$); see [loc. cit., Def. 16.1.1 and Prop. 16.1.2(a)] for details.
Let $V \subseteq Y$ be an open such that $f$ induces an isomorphism $f^{-1}(V) \stackrel\sim\to V$. Let $U = f^{-1}(V)$, $Z = X \setminus U$, and $W = Y \setminus V$. I claim that the cycle $\varepsilon := \Gamma^\top \circ \Gamma - \Delta_X$ is supported on $Z \times Z$. By the short exact sequence of [loc. cit., Prop. 1.8], it suffices to show that the restriction of $\varepsilon$ to $S := X \times X \setminus Z \times Z = X \times U \cup U \times X$ is zero. Let $$T := (X \times Y \times U) \cup (U \times Y \times X) = \pi_{13}^{-1} (S).$$ Then the restriction of $\varepsilon$ to $S$ is the pushforward along $\pi_{13}$ of $$\left.\left(\pi_{12}^* \Gamma \cdot \pi_{23}^* \Gamma^\top\right)\right|_T.\tag{1}$$ We can compute the latter as $$\left.\left(\pi_{12}^* \Gamma\right)\right|_T \cdot \left.\left(\pi_{23}^* \Gamma^\top\right)\right|_T.$$ This is a proper intersection, and the intersection is equal to the 'diagonal' $$\left\{(a,b,c) \in U \times V \times U\ \big|\ f(a)=b=f(c)\right\}.$$ Indeed, the intersection agrees with this set on both $X \times Y \times U$ and $U \times Y \times X$.
Then the pushforward of (1) is $\Delta_U$. Hence $\varepsilon$ vanishes on $S$, so it is supported on $Z \times Z$. In particular, the projections $\pi_{1,*} \varepsilon, \pi_{2,*} \varepsilon \in CH_*(X)$ are supported on $Z$.
The punchline is that $\varepsilon$ acts as the identity on $CH_0(X)$ by the moving lemma. Indeed, any $0$-cycle $a$ on a smooth variety can be moved away from $Z$, so the intersection $\pi_1^* a \cdot \varepsilon$ is zero.
This proves that $f^* \circ f_*$ is the identity, and by symmetry the same holds for $f_* \circ f^*$. $\square$
Remark. It is even true that $CH_0$ is a stable birational invariant: if $X$ and $Y$ are smooth and proper varieties with $X \times \mathbb P^n \stackrel\sim\dashrightarrow Y \times \mathbb P^m$, then $CH_0(X) \cong CH_0(Y)$. (The only thing left to prove is that $CH_0(X) \cong CH_0(X \times \mathbb P^1)$, since we already know the result for birational varieties.)
Remark. The reason Fulton only proves the theorem for algebraically closed fields is that Fulton only proves the moving lemma over those fields. For a proof of the moving lemma over arbitrary fields, see Roberts's appendix to the Oslo 1970 Algebraic Geometry proceedings [2] (finite fields are addressed separately).
References:
[1] Fulton, William. Intersection theory (second edition). Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge. Springer-Verlag, Berlin, 1998. ISBN: 3-540-62046-X; 0-387-98549-2. MR1644323
[2] Roberts, Joel. Chow's moving lemma. Appendix 2 to: "Motives" by Steven L. Kleiman. Algebraic geometry, Oslo 1970 (Proc. Fifth Nordic Summer School in Math.), pp. 89--96. Wolters-Noordhoff, Groningen, 1972. MR0382269
Using Jason Starr's comment I was able (I think) to figure out the case of $\pi_1(X)^0$. For anyone who stumbles across this with the same question in mind I add a sketch of the proof in an answer. For the experts, if you feel like leaving a comment whether my reasoning is correct, that would be great.
Let $\phi: X \rightarrow Y$ be a rational map, i.e. defined on some open $U \subset X$. Because $X$ is normal, $Z := X \backslash U$ has codimension at least two. This means the dimenson of the ring $\mathcal{O}_{X,z}$ is at least two for each point $z \in Z$. But then this local ring is pure (SGA 2, Exp. X, Thm 3.4), which in turn implies the couple $(X,Z)$ is pure (SGA 2, Exp. X, prop. 3.3), so that the categories $\operatorname{FEt}(X)$ and $\operatorname{FEt}(U)$ are equivalent.
Since $X$ and $Y$ are birational, we can find an open $V \subset Y$ such that $U$ is isomorphic to $V$ and in the same way we show $V$ and $Y$ have the same \'etale coverings. This shows $\pi_1(X) = \pi_1(U) = \pi_1(V) = \pi_1(Y)$ as desired, which proves in particular what I wanted to show.