If we replace projective variety with algebraic variety in the statement of the Weil conjectures what happens? To me it seems the statement still makes sense. But is it still true?

Correctly restated, the conjectures hold for any variety $V$ (not necessarily complete or nonsingular) over a finite field $k$. Dwork proved that the zeta function $Z(V,t)$ of $V$ is a rational function of $t$. Grothendieck (et al.) expressed $Z(V,t)$ as the alternating product of the characteristic polynomials of the Frobenius map $F$ acting on the etale cohomology groups of $V$ with compact support. Deligne showed (Weil II) that for each positive integer $i$ and each eigenvalue $a$ of $F$ acting on the $i$th etale cohomology group of $V$ with compact support, there exists an integer $j\leq i$ such that all the complex conjugates of $a$ have absolute value $q^{j/2}$ where $q=k$. When $V$ is nonsingular and complete, these statements together with Poincare duality, give Weil's original conjectures. 


As usually stated, most of the statements break. Try looking at $\mathbb{A}^1$ minus a point. There are still a lot of interesting things to say about the relationship between cohomology and point counting (the supertrace of the Frobenius on the compactly supported cohomology still gives you the number of points), but things like the functional equation and Riemann hypothesis depend heavily on Poincare duality, which, of course, fails pretty badly. 

