Do instruments get out of tune when you place them near a radiator when the radiator is hot?
If so, why?
Also, is there anything else that happens when you place an instrument near a hot radiator?
It depends on the instrument. Digital electronic instruments are probably OK.
String instruments, especially those with steel strings, will be seriously affected:
I suspect woodwind instruments are less affected. Brass instruments will expand when they get hot. Analogue electronic components will change their tolerances depending on temperature.
Adding to slim's answer, I've seen bridges pulled off acoustic guitars,strings break when not being played, bodies split because the wood has dried out,and necks spiral for the same reason.Some salvageable, some terminal.Apart from causing extra tuning problems, just don't do it. There should be a health warning issued with every new guitar !
One of my pianos has a heater in it, and a receptacle to keep water inside. Maybe not necessary in G.B., but other countries round the world.
Apart from tuning and condensation problems, brass and woodwind don't mind changes in temperature and humidity as much.
The humidity (or lack thereof) is a huge part of the equation here. Humidity control is essential for any fine instrument. Proper humidity keeps everything pliable, which in turn allows it to cope with fluctuating temperatures that are standard with winter and indoor heat. In my experience, it is the cheaper instruments (for example cheap acoustic guitars, and "student" upright bass, violin, etc) that are more resistant to adverse conditions; probably because they are made of plywood and resin; very durable materials. Finer instruments are made out of more delicate materials and should be treated accordingly. But any instrument, like any person, would prefer more favorable conditions. I would say, if you are not comfortable in a spot, your instrument probably isn't either.
I learned this the hard way years ago when my Lowden O10 emitted a "pop" thru its case one morning when the heat came on after a particularly cold night. I opened the case to find the bridge ripped off from the tension of the strings. I was absolutely devastated. Fortunately, it was fixed expertly and I've learned my lesson. I would never let any instrument I cared about go through winter without a Dampit in the sound hole and a humidifier in the room.