wiring furnace blower to turn on when make up air damper is powered to open
I understand you probably have a tight house and when your exhaust fan is on you want the furnace to provide fresh air to keep the smoke out of the house. 
I have used the power to the fan  to pull in a relay.  The relay and the furnace transformer to power the call for the fan just like if the thermostat was in fan only, this way if the call for heat or cooling is active when you are cooking there will be no problems with the controls. You will need to run a piece of Romex to a box in the area of the thermostat I have done this in the attic the hardest part was fishing the wires up the walls from the thermostat. I did this on the last house tract we built as one of the features of our buildings.

Edit for 24v operation after re reading.

Be cautious using power from separate systems , I re read the question and you could use a Separate 24 v relay that is powered from the damper 24 v to create a signal from your thermostat to your fan control signal.
then you would not need to tap the fan supply, when there is 24 v for the damper that same 24v can be used to fire a relay (think electronic switch) when the relay fires or pulls in it can send the signal to the fan the same as your thermostat so it won’t matter if the furnace is running or not.
when the 24v is there the relay turns on the fan. 
I would caution from Using the 24 from another source can cause many problems including damage to your thermostat and or furnace control board , the fan is usually the green wire or G terminal at the stat or control board, the red wire goes to the R or V terminal this is your hot  if you jumper red to green the fan should start , now add the relay on the red Common wire for the relay and and the green to the NO (normally open) contacts for the relay then run some door bell wire or better from the relay coil a1 and a2 or input and the other end to the damper where the pressure switch powers it or come from the pressure switch itself. If your 24 v comes from the furnace it is much easier just take the hot to the green terminal as the common or c wire should be the same, if not powered by the furnace use a relay as I have seen folks blow up 800$ control boards &amp; thermostats because they had 2 different transformers.