Latency is generally negligible for IIR filters. If you're aiming for lowest possible overall latency, go with IIR.
FIR filters can have pretty high latency. It depends on the length of the filter, which in turn will depend on the frequency range on which it operates (lower = longer) and the Q (higher = longer).
For room correction, you'll be targeting lower frequencies, so FIR filter lengths are likely going to need to be quite high.
Are you planning to do the room correction yourself, or use something that automates it? If the former, you may have more control over striking a balance between accuracy and speed.
In any case, IME there is rarely a need for FIR filters for room correction, as most of the phenomena you're correcting will likely be more or less minimum-phase.