How to connect c-wire to Burnham Series 2 boiler?

Tap the white wire in the control harness

If you look closely at the wiring diagram for your boiler -- there is a 9 pin connector with a harness connected to it that connects to the thermostat wiring junction, the gas valve, and the limit circuit. The white wire in this harness, while technically the gas valve common or MV/PV terminal, is a 24VAC common (C) wire, and can be tapped to serve other needs in the system. In fact, in the wiring diagram, it is shown running to the limit circuit extension plug as well for use by low-water cutoff or equivalent safety devices.

So, with power to the boiler off, we can cut the white wire bundled with the two blue wires in the harness to that extension connector and splice both of the cut ends together with a wirenut, giving us a bundle we can splice our thermostat C wire into. Install your Ecobee, turn the boiler back on, and enjoy!

As an aside, the reference in the control manual to not putting "auxiliary controls or limits" in the grounded leg means that no limit switches or control switches can be installed in a way that breaks the 24VAC return path from the control board to the transformer -- it is not a concern here.