Every closed immersion is a finite morphism. Therefore, restriction of a finite morphism to a closed subset is always a finite morphism itself. Can you give an example of quasi-projective varieties $X\subset Y$, $Z$ and a finite morphism $f:Y\to Z$ such that restriction $f:X\to f(X)$ is not finite? Same with Y -- projective?
PS. Sorry the original version of this question was hilariously stupid.