I have a 35 year old Whirlpool refrigerator. The compressor is too hot to touch. My finger will burn if it is touching for a full second. However, the refrigerator and freezer are fine temperature-wise and the fan is running. The compressor is obviously running (I feel vibrations before it got too hot to touch.) I've had it on for the last hour and it has not cycled off but I've also had it completely off for 4 days because I'm afraid of a fire. The vinyl floor under the compressor has been discolored (I assume due to the heat). I have also cleaned the coils under the unit. What can the problem be? How do you fix it? Is there some other place I should try to clean? Is it time for a new one?

Old compressors do get hot -- very hot. The problem is efficiency. They're not very efficient, and age doesn't help, and they lose a lot of energy as waste heat. I think you're absolutely correct about the vinyl underneath the compressor.

How to fix it? You could replace the entire compressor. That wouldn't seem very cost-effective. The claim is made that the newer refrigerators will pay for themselves in energy savings in 3 years (I haven't evaluated this, so I don't know for sure). The government claims that energy costs for refrigerators made after 1993 are cut in half. I've been trying to get my mother-in-law (95 years old) to replace her 1952 vintage refrigerator that runs constantly and doesn't get cold enough to freeze ice. She still refuses. :(

Energy Star Refrigerators and Freezers

The compressor will get quite warm. Cleaning the coils certainly should help. If the refrigerator has been off for 4 days it will take more than 1 hour to cool the inside down, probably 8 to 12 hours. The compressor has a thermal switch to protect it if it gets too hot.