Should I bypass a thermal cut-off switch on my oil furnace?
I would really want to know why the switch failed twice, and running the unit without all safety devices seems like a bad plan.  I'd find a new tech (don't use the other guy again) to diagnose the actual problem and install a new switch correctly.

The tech may have never seen a thermal cut-off switch do anything and decided it wasn't needed, but that's not really something that should be done.  Your neighbor is right - get it fixed properly.