By demand I expand a little on my answer. The holomorphic Lefschetz fixed point formula (aka the Woods-Hole formula) considers an endomorphism $f\colon M \to M$ of a smooth and compact complex manifold $M$ (or proper smooth algebraic variety) with only isolated fixed points which are also assumed to be non-degenerate (i.e., the tangent map of $f$ at a fixed point does not have eigenvalue $1$. Then the alternating trace of the action of $f$ on $H^*(M,\mathcal O_M)$ is equal to a sum over the fixed points $p$ of $1/det(1-T_p(f)$. In particular if there are no fixed points the alternating trace is equal to $0$. However, in the case when also $H^i(M,\mathcal O_M)=0$ for $i>0$ then the alternating trace is equal to $1$ so the assumption that there are no fixed points gives a contradiction. Note, that the dimension of $H^i(M,\mathcal O_M)$ is just $h^{0,i}$ so that an assumption that the Hodge numbers vanish off the diagonal gives the required vanishing by a good margin. Furthermore, the vanishing of just $h^{0,i}$ for $i>0$ is much much weaker, it is for instance a birational condition whereas blowing up a smooth curve of genus $>0$ in a variety of dimension at least 3 always give off diagonal Hodge numbers.
As for the question of whether a birational involution of $\mathbb C^n$ always has a fixed point this seems trickier. It is true that the involution can be made to act regularly on a smooth and proper model and hence by the above has a fixed point. It is not clear however that the fixed point will map to a point of $\mathbb C^n$ as $\mathbb C^n$ is not proper.