So the ISS uses small thrusters on the Soyuz (usually) to perform occasional reboost burns to raise its orbit. This goes through, from what I can tell, around 3,500 kilos of fuel per year. Would it not be far more efficient to use ion drives to counter the drag instead? It would require far less fuel and I don't understand what the reason is for not doing it. The thrust can easily be higher than the force of atmospheric drag on the station, from what I've read. It could simply fire on and off periodically, and of course other stronger thrusters would be kept for evasive maneuvers and whatnot. Is there some obvious problem I'm missing?
To add to the other answers - low thrust means long "burn" time, ion engines are used for continuous thrust for days. And that might be a problem for ISS which is used as a zero-G (well, micro-G) experiments laboratory, because even the ion engine-induced acceleration might mess with these.
Ion propulsion uses accelerated ions to produce thrust, AKA Coulomb's Law of electrostatic attraction/repulsion. A neutral substance is ionised to produce the ions.
Due to this, only a neutral gas (usually Xenon), and a power source for the charged electrodes (and neutraliser) are needed. Depending on size, the power consumption of current ion thrusters use 2-3kW. The Deep Space 1 used a 2.3kW NSTAR thruster.
However, thrust forces are very low compared to any liquid fuel or solid fuel rocket engine, being anywhere between 30 and 300 millinewtons for ion thrusters, up to 92mN for the NSTAR. Due to the low mass of fuel required, independantly the engines have a reasonable thrust to weight ratio, so for small probes this is an ideal solution, resulting in incredible amounts of Delta V due to the Ion Thrusters high efficiency and specific impulse rating.
For your original question, Ion engines would simply not be practical currently for an effective propulsion method for the ISS. Primarily due to the huge mass of the ISS, many, many Ion thrusters would be needed, being extremely costly and even then the thrust to weight ratio would be extremely low. Furthermore, the power consumption would also be far too high, for the quantity of thrusters needed, and their increased burn time due to the low thrust to weight ratio.
In short: Ion thrusters are ideal for low mass satellites or probes, but until technology advances are made they are too small and power demanding for high mass applications.
I will run some numbers later for thrust to power input ratios.
I believe, at least one incarnation of NASA's LOPG or previously Deep Space Gateway plans use electric propulsion to adjust orbits, however I believe this may well be plasma rather than Ion.
@Hobbes' great answer and the question there as well show that both NASA and Ad Astra felt since 2008 that high power ion propulsion was both technically and practically worthy of consideration and testing for at least some of the routine orbit-raising maneuvers required for regular compensation of the constant drag at about 400 km altitude.
However the plan was ultimately cancelled in 2015.
The conventional propulsion situation in 2014 is outlined by several answers to the question What propulsion methods does the ISS use for station-keeping? but the situation has possibly evolved since then.
It is not clear to me if ion propulsion could potentially handle regular orbit-raising. However, should they need MORE thrust than that for changing orbits to move out to the way of space junk, or possibly for some situations requiring quick phasing related to meeting spacecraft from Earth or sending them back there, they would likely still have to rely on chemical engines to do it quickly enough.
So while the use of ion engines could reduce the weight of conventional propellant regularly shipped from Earth, it would not completely replace conventional engines on the ISS.