My fan belt (or something very near it) sometimes squeaks, but only when the engine is at idle, and warm. Start up in the morning - nothing.

The fact that it's only when warm / at idle - does that suggest anything? Any good way to test for what it could be? I could jump to "new belt" conclusion, but I hate replacing things before being pretty sure what the issue is.

 

I would warm up the engine then shut it off.
Pull off the belt and spin the pulleys to see if any are making noise, idler or tensioner pulleys bearings do start to fail and make noise

Each time you hear a Fan belt squeal that noise comes from the fan belt slipping on a pulley(assuming all pulley bearings are good now)
So each time you hear the noise the fan belt has Glazed(shined up) an area on that pulley as it slipped.
So when it hits the Glazed area again.............easier to slip, this is why squealing just gets worse over time, more and more glazing of pulleys

People often put on a new fan belt and wonder why it starts to squeal pretty quickly.......
Well it is slipping on the Glazing on the pulleys

So if you will replace the belt then get some scotchbrite or emory cloth and rough up the pulley surfaces, get rid of the glazed areas.
Painting pulley surfaces is good to, friction is your friend where belts and pulleys are concerned, smooth is bad