It is not your filter burning. But the filter may be the cause of the motor heating up. If it is plugged then it puts a strain on the motor because it can't pull air through it. It would be like as if you put a bandana over your mouth and then tried to jog the same distance at the same pace that you normaly do. Your body would be straining pretty hard. Eventually it would over heat and either you stop or your heart would.
Check the filter, and if needed, change the filter.
If the filter is ok then you probably have a motor going out. The smell should be a sharp sweet smell. If this is what you have been smelling then the motor is the culpret.
You say that the motor runs and then stops. That should be when you smell the smell. If the smell is more like a burnt rubber smell then your fan may be jammed. This is easy enough to diagnose. It smells like burnt rubber and there is no air being sent to the registers (where the heat and a/c come out of). If this is the case you need to find out why the fan is not turning.
Most likely it is going to be the motor. If you are handy with simple hand tools this is really not that hard a part to change out (most times, but not all). About the hardest part is hooking up the wires. If you make a diagram of how they come off then it is easy to put them back on the new motor. Be sure that the new motor is the same as the old one. Sometimes they have an after market motor that is diffrent than the OEM one. If there is no instalation papers with the new motor then you will need to call the retailer and ask which wire goes where. If they dont know take it back and get a diffrent one.
If you are not mechaicly inclined then a handy person is not that expensive to hire to do the change out. I charge anywhere from 50.00 to 100.00 depending on the complexity of the heater unit. This includes the diagnosis. If you can see the motor. Do the nose test. The motor should smell just like the smell you have been smelling.
I hope this has answered your question. If you are in any way unsatisfied with my answer DO NOT ACCEPT IT. Reply with what you wish to know and I will answer you promptly. If this has answered your question then please be sure to click the accept button.
Best regards; The Home Smithy