Probably, but not for a long time.
JWST is fueled with hydrazine thrusters (similar to RCS) and of course, it only has so much fuel. This isn't Spaceflight Simulator; we can't have infinite fuel (please excuse my gaming humor). While the probability that JWST will run out of fuel is high, the chances it happens before they run out of coolant is very low. The thrusters will likely not be an issue at all. Another factor on Webb's side is that the Ariane 5 is a very, very good and accurate rocket, like a European version of Atlas V (makes me think that 5 must be the lucky number of rockets, Ariane 5, Atlas 5, Saturn 5) and so, the injection burns were really precise.
As astrosnapper also mentioned, the End-of-Life maneuver (Or as I call it, the space coffin) will require fuel. For example, geostationary satellites are usually moved into a very high graveyard orbit at the end of their mission. I can see something similar with JWST, except for the fact that JWST is about as high as it gets already.
Basically, when you put this all together, no, thruster fuel likely will not be an issue.