I changed the fuel pump and also a relief valve that was leaking. Primed and bled the system, it fired right up. Let it run through one cycle, towards the end of the cycle it started doing the same thing was doing before.
I opened up the line before the fuel filter and found that it was dry. I opened the fuel filter and it was only about half full. I opened up the line on the tank side and fuel came pouring out. So... I disconnected both ends of the fuel line, hooked up a line from my air compressor and let it rip. A whole bunch of crap came flying out the other end. Black sludge. So I guess my initial thought that there wasn't any restriction in the fuel line was wrong!
After doing that, I put the lines back together, bled it again and fired it up. It's been running fine ever since. Plus I've got a new fuel pump, all new filters, a new relief valve, etc...
Oh, BTW my fuel tank is in the basement, about 10 feet from the burner. So the heat tape wasn't necessary.