Ecobee pro rebooting

Get out your trusty multimeter. You should read 24 VAC from RC and RH to C. You may need a jumper from RC to RH.

no need for Yellow or Green

Does your Ecobee have an equipment configuation that allows you to choose "RC only"?

Maybe this will help

https://support.ecobee.com/s/articles/3-wire-heat-only-thermostat-R-G-W

What is the model number of the Ecobee? There is a reason for the problem, it could be in the initial setup of the thermostat, It could be that RC and RH need to be jumped together, I would need to read the manual. I can find the manual online if I had the exact model number ot your thermostat.

In most smart thermostats, R or Rc is the terminal that goes to power up the onboard computer. C is the return path from that onboard computer to the power source. So you need to connect RC and C to the same transformer to get the thermostat to boot up properly.

Now depending on how the manufacturer sets up the internal components, you may or may not need RH in order to connect the W on the thermostat to the burner control(s) I'm not sure how you thermostat is designed, so the model number of that thermostat will help me locate the technical documents online.

OR

see if the RC to RH jumper wire does the trick.

If it was ever shorted… poof. However, if you are getting that 28 VAC at the terminals for the Ecobee, the transformer — and wiring is at least intact. However — and this is hard to check — if there is a bad connection in any of the wires, or if one of the wires has been chewed to the point where it is hanging by a thread, that can introduce enough resistance to drop the voltage low enough to cause problems. Check all the connections for being clean and tight.

Look again. there is a diagram attached.

@EdTheHeaterMan

Can I just use an external power supply/ outlet like this to keep it from rebooting?

That may work however, that will not have a relay onboard like the Beckett 5915U that you already own. And that Beckett 5915U already has the same transformer the Ecobee power extender kit has. You can male your existing 59150U work exactly like the Ecobee kit this way:

You can just plug it into a wall outlet this way. or you can hard wire it under the control. What ever you are more comfortable with.

And according the the Ecobee Kit diagram you may not need the Y1 on the Ecobee thermostat connected to a relay. I just know that some other brands need a relay connected to sense the resistance of the coil. Apparently your Ecobee does not.

where are you getting the 120 VAC for the transformer from? is it the Limit or L1 to the primary control? If you do not have a hot LI all the time, then every time the pressuretrol or the LWCO cuts off the burner, it also cuts off the 120 VAC to the transformer which caused the 24 VAC to cut-out. That may be your problem. This is the wiring diagram for your boiler.

You have the Beckett GeniSys so we are using the bottom burner diagram. there should be a plug to connect the burner to the boiler, but it appears that someone replaced that wire with MC armored cable wire. If you do not have 3 conductors + Ground (4 wires total) then you may not be able to do this yourself. You need to have a red white black and green wire from the burner to the boiler. The 24 VAC transformer primary side (120V) must be connected to the black wire that goes to L1 on the Primary and the white wire that goes to L2 on the primary. If you don't have the red wire to go to Limit on the primary, then your burner will not operate properly. You MUST have the RED limit wire. If you do not, then that is your problem.

If you look at the boiler wire diagram on the left side you can clearly see that there is a HOT always wire labeled PK that goes to the plug that connects to the Black wire in the burner side of the plug. If you do not have that always hot wire to the burner then you need to get one there and connect it to the L1 on the primary control and use it for the transformer. (you may need to disconnect a jumper or wire nut that connects both L1 and Limit together in the oil burner control box.)

This can be confusing so take your time to look closely at the diagram to make sure you have all the wires you need. Look at each wire one at a time to see that they are connected properly.

This is your problem:

Sorry for the late response. Been stuck at work and haven't had a chance to stop by my grandmathers house to check the wiring.

So base on the diagram and the explanation.... The MC armored cable wire has black, white, green and a red wire (but the red wire is not plugged in, it's cut off see pictures).

The transformer is wired to the black and white wire from the MC armored cable wire.

The black wire is connected to the limit circuit on the boiler. So whenever the LWCO or the Pressure control opens the circuit (the way it is supposed to) the 24 VAC is also interrupted. The Ecobee understands that as an error. You will need  to connect that Red Wire on both sides to solve your problem. The way it is you only have 2 conductors and a Ground.  Like this:

You need 3 conductor and a Ground like this: