My furnace thumps like someone is hitting the sides of the case. Doesn't start making the noise until the burner has operated for a while, and stops after the burner shuts down, but , before the blower shuts down. The flame looks good,and the noise is distant sounding when the furnace covers are removed.I have been unable to narrow the sound location down by looking while it is making the noise. The furnace does not seem to overheat, as I can touch all over while it is running and not find any spots warmer or colder to touch. Could use some ideas

2 possible reasons. #1 is that a piece of ductwork has come loose and is flapping around or #2 You are getting an "oil can effect". Sheet metal is expanding and contracting-the clue is your comment about heating and cooling-and bumps one way and then the other. Usually found in a large duct. Keep looking for the noise. When you find the spot, putting an X crease on the duct with a chisel will eliminate the issue. Not a malfunction, fairly common so keep the heat on and stay warm.

I have the same problem - its been going on for two winters now. So far the house has exploded so I guess we're safe enough. It is unnerving at times.

Correction - our house Has Not exploded