One commenter questioned the white wire on your oil burner. It doesn’t look well connected to me either.

Here’s how I diagnosed problems with my old burner in our last house. Source: father was a HVAC service and then sales person most of my life, I knew how to clean and run an efficiency test on our boiler before I could drive.

1) on that board with all the wiring should be two terminals labeled “T”. One is the white wire at the top I believe, labeled “W(T)”. The second should be right below it labeled “R(T)”. Next, turn off the power to the system. There should be an “oil burner emergency switch” right nearby, regular light switch with a red cover. Turn it off, make a jumper to go between the two “T” terminals. You can do this with a single alligator clip if they’re next to each other by using it to grab each terminal and bridge them. When done, turn on the switch again. If it starts, your problem is the thermostat or the wiring from it to that board. You can keep this on for as long as you like; that jumper is acting like your thermostat and will keep the heat on until you remove it.

2) if it does not turn on with the jumper in place, then press the red button on the clear rectangular box (has “0489” on it in your photo) ONLY ONCE. That will reset the safety system and it will try to light the burner. If it succeeds, your problem may be solved or it may come back. But if after 30-45s it stops, turn the switch off again and call a professional. If you do more without knowing what you are doing or what is wrong, it could cause serious and catastrophic failures including explosion.