Probably not the issue, but first check is to check your oil level. I recently was called out to a similar issue that turned out to be the oil was robbed from the tank.
Apart from that, it is a burner issue and will probably need a service call. It could be a number of causes, such as faulty photocell, dirty photocell, faulty fuel pump, faulty solenoid coil, etc. Hard to diagnose remotely without seeing and hearing the problem and being able to test the various components. It will not be a thermostat problem.
A common fault is the photocell which is an inexpensive part to replace, but will probably still need a service call. If it has not been service for a while it maybe worth getting this done at the same time and it may work out cheaper.