Suppose I have a quasiprojective variety $X$ and a finite surjective map
$$f: X \rightarrow Y$$
to a scheme $Y$. Is it true that $Y$ is quasiprojective as well? It seems like the answer could be no, but I don't know enough examples of non-projective schemes.



Finite map from quasi-projective variety
This isn't true in general. 


For example, see Section 6 of *Conducteur, Descente et Pincement* by D. Ferrand. There Ferrand gives an example of a non-normal proper variety $Y$ whose normalization is projective. 


If I recall correctly, many examples of proper non-projective schemes have finite maps from projective ones.


