Will the ions in salt bridge be used up?
The ions in the salt bridge are meant as charge carriers, and using a suitable diaphragm intends to keep the mass transport pretty low.  

Nethertheless, some of the electrolyte (like $\ce{KCl}$, $\ce{KNO3}$, $\ce{Na2SO4}$ or $\ce{Bu4N+ ClO4-}$ to mention a few) does enter the other half cell.  This is why, as one criterion, the electrolyte in the salt bridge is choosen to not interfer with the electrochemical reaction you want to characterize (analysis like cyclic voltammetry) or execute (electrochemical synthesis).  Likewise for solvents used in cyclic voltammetry, there is an electrochemical window, the intervall of electrode potentials where it is safe to use them so that they are not oxidized, or reduced, respectively.

And if the electrolyte is consumed and charge carriers are absent, provided the electrical potential applied is not altered, yes, the electrical current drops to 0.