The pump may be partially clogged with debris, or on the verge of failing. It's probably time to call a service tech for a 'tune up'.

Randy's explanation is most likely, but I will relate a similar situation I had.

I was having issues sort of like you describe where if my oil-fired hot water heater was running and the furnace switched on, there was insufficient oil flow to feed both, so the furnace would kick off on safety, and I'd have to hit reset. This started happening 8 years after initial installation. Even after purging the oil lines multiple times and replacing filters, it was recurring. My oil tech told me it was probably because at some point in the past, I had gotten some bad oil, and there was junk in my tank. He replumbed the oil to draw from the top of the tank. No problems since.