Upgrading 2-wire to Ecebee3
The Beckett GeniSys™ Advanced Burner Control is used to control the operation of the furnace, and does not appear to supply an external C terminal.  However, you may be able to power the Ecobee using an additional 24Vac transformer.

If you have an A/C system that the Ecobee is controlling, you can power the Ecobee using the transformer in the A/C system.  Just connect the R wire from the A/C to Rc, and run a C wire from the A/C to the C terminal on the Ecobee.

If you're not controlling an A/C system, you'll have to use an external transformer (or if there's one in the furnace).  If there's a transformer in the furnace (in the control box with the CYGR terminals), then you should be able to use that (assuming it's connected to power on the primary side).  Simply run a (new or additional) thermostat cable from the furnace up to the thermostat. Then wire it as follows:


Remove any jumpers between R and Rc on the Ecobee.
Connect a wire from the R terminal on the control box in the furnace, to the Rc terminal on the Ecobee.
Connect a wire from the C terminal on the control box in the furnace, to the C terminal on the Ecobee.


Assuming the transformer in the furnace is powered, that should get you up and running.


  WARNING: Ecobee is an expensive piece of equipment, make sure you verify proper input voltages. You may also want to contact Ecobee support, to verify this is an accepted installation method.