I am trying to imagine
. If it’s basically an open cupboard with a shelf it shouldn’t get too warm. Boiler noise tends not to be too bad. And they do get used to things. Our hammy is in a room with a freezer that makes noises when it goes on and off. I’d be more concerned about restricting air flow to the boiler for safety concerns. Instruction books usually say they need a certain amount of space around them. But if it is say at floor level and the boiler well above it that would be ok - so the cage would be in a kind of alcove?
My only other thought is - is the cage big enough if it fits the width of an airing cupboard (although realise airing cupboards come in different sizes.
You need a cage at least 80cm by 50 cm and no more than 46 to 50 cm tall for a Syrian or 38cm ish tall for a dwarf hamster.
80 x 50 or 75 x 45 is a good size for a dwarf hamster. Most people go a bit bigger for Syrians with something like the Hamster Heaven being the minimum suitable really.