Why would the turn signal flasher unit stay hot?
Please verify that the CORRECT flasher/relay for the vehicle in question is in fact fitted. The wrong one could cause what you are describing with nothing else actually wrong. Substitute a known good flasher/relay of the correct type for the overheating one and see if the problem goes away. If the flasher/relay is at fault there will be no further sign of the problem, but if it does not then vehicle electrical system is causing the problem.
Unplug and/or substitute a known good hazard switch for the one in the vehicle (the turnsignal power lines usually go thru the hazzard switch) and see if the overheat stops.
Does the vehicle have a tow package electrical adapter (like U-haul) installed? These contain diodes inside that can short and cause low level load related problems.
Check the ground connections between the battery and vehicle body. Do this with a voltmeter, there should be less than 0.10v voltage drop across any ground (one probe on the battery post and the other on the body where the cable attaches) with a load on the cable (meaning with power flowing thru it). I have seen a bad ground cause a power feedback thru unexpected pathways as the electicity tries to get back to the (badly grounded) battery post using the engine block ground cable.
With the preceeding in mind...
Do the turnsignals have to be used befor it heats up or will it heat up just from turning on the key? Do the turn signals/parking lights ALL work properly? By that I mean do they come on only when they should, and go off only when they should? If some lights are coming on when they should not, and any bulbs were recently replaced, check to make sure the correct replacement bulb was fitted. I have seen unusual operation when a single filament bulb was fitted to a dual filament socket (yes, it will fit, and the single filament bulb-contact bridges the dual filament contacts in the socket creating a "short" between turnsignal and tail light circuits).
Finally, with the electrical system in the problem mode (IE turned on, etc as needed to create the overheat condition) but the turnsignals (and hazard switch) turned off, disconnect or pull the flasher/relay out of it's socket and watch for the flash of a spark. You may need to use a darkened environment such as a garage with the door down or outside at night to see the flash. If you DO see a flash from a spark then there is a definite load on the flasher/relay when there should not be. Try to see if any lights or other loads go out when the relay/flasher is unplugged and if you do see something change that may be the issue or related to the issue at cause.