I'm sure there are various notification devices out there that can let her know if the temperature drops, as opposed to having to rely on the neighbor.
Now for hitting the red button, I'm gonna say no. That is a reset and usually flips when an error or fault condition occurs, thus requiring manual intervention.
I would be hesitant to push that button without being onsite. If you keep hitting that reset button and the burner isn't firing you can flood the combustion chamber with oil. Then when the burner does ignite, the pipes freezing will be the least of her concerns!
Keeping the burner serviced, nozzle cleaned, filter changed and oil in the tank should keep everything running smoothly. It does happen when the reset needs to be pressed, but that should not be a normal occurance.