1989 Bally Mousin' Around - right flipper coil is overheating and flipper stays up (or is that the other way around). Where do I start?

I would check to ensure your coil is correct type for machine and EOS switch is not dirty / broken

Edit -

Futher does the flipper itself move freely?  Ie just by hand.  Make sure it isn't binding on the bush as well....

I would also check diodes on the coil on question too....

If you have a DMM - it is possible that a winding has broken on the coil too....