I wanted to share an experience I had a few weeks ago, maybe it'd help somebody else out.
I was working from home, at the dining room table. My oil burner is in the basement, right below the dining room. I would hear it turn on, then after about 30 seconds the sound would change from the normal "low roar" sound to a weaker sound. It would switch back from this sound to the normal sound, then it'd shut off after another 10 seconds or so.
So... I called my neighbor, who has some HVAC experience. He checked it out and thought that the pump wasn't getting oil so we popped open the filter. It was full of gunk. Aha! we thought. Changed the filter, fired up the burner.
Then we checked out the burner nozzle, found that to be clogged. Aha! we thought, and changed that too. Fired up the burner, everything seemed to be ok.
The next morning I woke up and it was 58 degrees in the house. The burner had shut down overnight.
That night he came back over after work. When the sound of the burner changed we cracked open the fuel line (between the filter and the fuel pump) and found that there was no fuel. Aha! we said again, thinking that the fuel pump was bad. So we changed that. Fired it up, everything seemed to be ok. Then the overflow started leaking.
So we changed the relief valve on the overflow pipe. That fixed the leaking problem. After we fired it up again it started doing the same thing again... We cracked open the fuel line at the fuel tank, oil came pouring out. Opened the fuel line at the other end (at to the filter) and there was nothing there.
I brought down my compressor, connected one end to the fuel line by the fuel tank and put the other end into a jug. I forced some air through the line and a whole load of gunk came flying out. The fuel line must've been mostly closed up, and when the fuel pump started demanding more fuel the line just couldn't pass enough through the gunk to keep it running.
After changing the filter, nozzle and fuel pump it turns out that it was just a clogged fuel line. Oh yeah, after all of this we also found that the expansion tank was no good so that was replaced too.
Now I've got a burner that has mostly new parts that burns nice and clean, and runs great.
Next time your burner won't stay running, check to make sure you're getting enough fuel to the pump. If not, check your fuel line!