I have no experience with this, but have been fooling around with my thermostat recently.
From reading
http://s3.supplyhouse.com/manuals/1350893573345/84050_PROD_FILE.pdf , it looks like this aquastat doesn't require power to run. So, the only transformer involved is the one for the furnace. So, at least there's no question of different sources.
For my thermostat, this is a typical wiring diagram. It looks like R and G are the wires to run the fan.
View attachment 168555
Thinking out loud, if the thermostat is still connected to the existing system and calls for heat, it will close R and W, the furnace will start up and the fan will likely run when the furnace tells it to. If the boiler aquastat is connected to just R and G, it would seem that the fan would start every time the boiler came up to temp, regardless of what the thermostat says. So, if you are just wanting to remove heat from the boiler, as in, it doesn't cycle or have storage, then I guess you could do that-it would be like burning a wood stove, but all the rooms would heat up evenly. I wonder if there could be some scenario where both the furnace and the boiler are running at the same time-that could be something where you'd want a relay to lock out the furnace. The original heating system would take over when things cooled down.
If you did have storage, (I think cycling is considered bad from an air pollution point of view), I think it'd be a different story.
Maybe you should post what your plans are.
And again, I'm just thinking out loud.