Have you checked the voltage at the solinoid while it is turned on? It sounds to me like you have a low voltage at the solinoid, causing a higher than normal current
draw, thus causing the solinoid to heat up.
Quick way to check would be to run a seperate wire run direct from a battery
to the solinoid, and see if it heats up then. If it does not heat up, start checking the existing wiring
for bad connections. A bad connection will also be heating
up, potentially setting up a fire situation.
Pay close attention to the ground side of the wire run, this is usually wher you will find the problem.