Not an expert, aside from countless hours in kerbal space program, but I would guess the problem is more about the total mass and structural strength. The station has some kind of thrusters to counteract orbital decay, but they wouldn’t have nearly enough fuel to boost to a higher orbit. So another craft with lots of fuel would be needed to push the station.

Something low thrust like ion thrusters would probably take a very long time for something the mass of the ISS. And you can’t just burn continuously. Raising an orbit is a two step process: burn to raise one side of the orbit, then burn again to raise the other. These burns are most efficient when done at the lowest and highest points of the orbit, respectively. Too long of a single burn would waste precious fuel from being too far away from the optimal points. I would guess that it would take many, many orbits to raise the station into a permanent orbit.

A higher thrust engine pushing the station would solve that problem, but since it wasn’t designed to be pushed, I could see it being unable to withstand the stress. Plus, it might be difficult to thrust along the center of mass, causing it to tumble during the burn.

That’s just my layman explanation, anyway. I imagine it won’t be easy no matter what, and it may ultimately not be feasible at all. But I’d like to see more public discussion of preserving the station.