Tate showed that the functional equation for zeta functions of number fields can be proven with fourieranalytic methods on the adele ring. Can the same be done for zeta functions of varieties over finite fields?

This is done in Chapter 7.3 of Ramakrishnan and Valenza's Fourier Analysis on Number Fields (GTM 186) which, despite the title, describes in some details also the situation over function fields in one variable (so, for curves over finite fields). If you want to consider curves over some global field of positive characteristic (so, not a curve $C$ over $\mathbb{F}_p$, but may be a curve over the field of rational functions of the previous $C$), Keerthi's comment applies: in particular, a recent work by Fesenko (Analysis on Arithmetic Schemes 2) treats the global functional equation. 

