One potential problem that occurs to me - is that it must be possible
to open firedoors even after they've been subjected to fire (e.g. in a
rescue) - so there's rules about how firmly the intumescent seals
might cause the door to bind. Would the paint cause the door to bind
into the frame in the same way?

(In practice, I'm sure the fire service deal with everything in their
stride)

Often you see firedoors varnished - not sure if this is because
manufacturers tend to supply them this way, or some technical reason -
but it does help them stand out as firedoors, and would show up any
bodged-up alterations that might compromise them.