In theory, is it possible to make a logic gate that uses zero current?
Yes. You can make a gate that switches with zero current if you don't
mind waiting an infinite amount of time ;) Since current is change
in charge over change in time, as the change in time goes to infinity the
current goes to zero. Run your logic as slow as you can while meeting
your other system specifications. 

Your homework assignment for tonight is to read the "Thermodynamics of
Computing" chapter from "Feynman's Lectures on Computation" ;)