My Kenmore 721.626222 which I have had about 6 years, runs but does not heat. There is also a strange smell - something like a burning smell.

if smells like something then something is burning stop using answer about can it be fixed is unknown till diagnosed

GET IT CHECKED UP///LOOKS LIKE SOME THING IS WRING WITH UR TRANSFORMER INSIDE UR OVEN...
I WOULD RECOMEND NOT TO USE IT UNTIL THE PROBLEM HAVE BEEN FIXED...KEEP IT UNPLUGHT FROM UR POWERPOINT

This is most likely a defective magnetron, especially with the burning smell. There are two other parts that can be a problem, the high voltage capacitor and diode, all three work together to heat food. If the magnetron is the culprit, probably cheaper to replace the microwave oven.

You can get support and parts at the Kenmore Sears site
http://www.kenmore.com/shc/s/v_10154_12604_Kitchen?adCell=WH&adCell=WH

Possible that the fittings along the magnetron had come to some dampness that has resulted in the heating of the mountings that causes the burning of the mounting gaskets.
If will be best to remove the magnetron and refit it after cleaning it dry and making sure there are no food particles lodged inside that can cause this out of turn heating to happen.
please check also on the main capacitors and the power supply for any heating.
Cleaning the entire unit after swittching off power , cleaning the PCB and the contact leads must solve the issue. If there are any loose joints tighten them.

Is this oven still under warranty? If so, take advantage. If not, remove the power plug and the case. Short out, with a screwdriver, the large can-shaped capacitor with two terminals sticking out. Check the wiring and circuit board for signs of burning. If everything looks ok, remove the thin sheet metal wave guide that sits above or beside the microwave energy port into the cooking cavity. Under the sheet metal, there is a metal fan-like device called a 'stirrer' that has a plastic bearing. Remove this stirrer and smell the bearing. If it smells like the burning wire smell, replace the stirrer. You can buy it through 'Sears Parts' online.

More common then you think, i have gone through a couple microwaves in the last ten years, they both failed very similar to what you describe. One popped the breaker one day, reset the breaker and it worked again but had an odd smell to it. I took it apart and couldn't find anything burnt or melted so i tossed it and got a new one, i didn't want a microwave burning the house down! I would suggest unless your really tied to that microwave that you just replace it for safety sake. Four years or forty years old don't matter anymore, i had one microwave, LG brand, that died two months after the warranty expired and it would cost more to repair it then replace the thing!

YOUR MAGNETRON IS MOST LIKELY BAD. THIS IS THE UNIT THAT CREATES THE MICROWAVES.

Turn if off, and leave it alone, remove it, if you can remove the unit, take it in for service, something is wrong if you smell electrical, most people for some reason can smell and identify correctly electrical or at least insulation burning smells, which means something is burning internally, not safe, fire hazard, unit needs to be repaired by qualified technician or replaced with new unit, take no chances, nothing you can do to repair this, could be a shorting transformer, or shorting power cables, but to attempt to trouble shoot this with email is a long shot, and depends on your experience expertize and testing equipment.

I wish I could give you a short easy answer, but the hazard and your safety is more important than a microwave.

Will

A component of the high voltage section has shorted, possibly the diode. If it was the capacitor, it would normally blow th fuse.
If it is the high voltage transformer and you continue running it,
it should devolpe smoke in about one minute. Either way, a part
will have to be replaced to correct the situation.