1. subtract a loop of the 'quiet' room from the live input. This is simple but probably wouldn't work too well because, if there is any irregularity at all in the recording, this will create an audible pattern.

2. take the 'quiet' room recording and put it into a [sigmund~]. Keep the highest amplitude value in the loop for each frequency band and use this to set the strength of a notch filter for the respective band. I've got a bad feeling about this idea- like it'd work well with clear, isolated pitches, but probably not with noise (which is going to be the main thing you want to filter!)

You could try cancellation by phase inversion, bypassed as a threshold is met so as to pass "non background" noise, but it will be no more effective than a noise gate.... and will introduce nasty artefacts.

Noise cancelling headphones work by phase inversion, but the program material is known and so can bypass the function. With live microphones that is not an option.
David.

Also a +1 for @lacuna's suggestion, I tried out that multiband filter to prevent feedback when using an open mic. It's computationally expensive, but it works nicely.