Is the map $Q\sqcup Q\to Q$ of quasi-projectives finite? Suppose you have two quasi-projective varieties $Q$ over an algebraically closed field. Let $Q\sqcup Q$ be their coproduct, so the identity maps $Q\to Q$ induce a morphism $Q\sqcup Q\to Q$. Does anyone know if this map is finite or not? Thank you.

Yes that map is finite.   
Since finiteness is local on the target , you may assume that $Q=\operatorname {Spec}A$ is affine.
Your map is then dual  to the diagonal morphism $\Delta:A\to A\times A  :a\mapsto (a,a)$ which is clearly a finite morphism of rings, so that your original variety morphism $Q\sqcup Q\to Q$  is finite too.
Edit
The above is valid for a completely arbitrary variety (or scheme).
No hypothesis whatsoever (quasi-projective, algebraically closed base field,...) is needed.
