Why would a transformer for a heat pump/central air conditioning unit overheat or start smoking?

Wires inside may be touching or have lost the insulation that prevents short circuits... I would definitely have someone come look at it asap and shut it off till they arrive.

A transformer is just like any other electrical device. It could have just gone bad OR something in the equipment could have caused a overload. Call for service.

A transformer is just like any other electrical device. It could have just gone bad OR something in the equipment could have caused a overload. Call for service.

I can't afford an expert right now and it's freezing is there anything I can check or do?

Well if the transformer is defective/burnt you can replace it. In order to really check out the low voltage end of the system you will need special meters etc. Most homeowners do not have the proper tools to do the so you can roll the dice and replace the transformer and hope for the best.

Now if you have the meters and know how to use them you can check into the system further and make a better diagnosis. I here about the lack of funds situations every day that people are faced with but that does not make it any easier or harder to tell people what they should do. I am in the same position. When my car breaks I can no longer work on it since it is all about computers today so if I want to use my car I have to pay someone to fix it. It is all relevant. You have to devote money to the basics first. Food and shelter are first and that means you will need to heat that shelter.

If you have the talent to replace the transformer that is a good thing. If you want to try and protect the new transformer since you have no idea if another problem caused the transformer to go out to start with then I suggest you stop by Radio Shack and get a fuse holder and a few 3 amp fuses and splice the fuse holder into the electrical line of the transformer. This way if there is another problem the fuse will blow and save the transformer. If the fuse blows then you will have to start checking more items that may be causing the problem but here again you will need some electrical knowledge because this work has to be done ON Site and not on a message board. In other words I cannot test it from here.

If you have problems replacing the transformer post back. But they are simple two 120 volt wires in and two 24 volt wires out. Pretty simple.