Furnace not heating after changing thermostat
If your furnace and a/c are very old they may have separate power supplies. Old school heat/cool stats worked like this Rh(24v power) closes to W(heat signal) on a call for heat, that's the heating circuit.
Rc(24v power) closes to G(fan) and Y(air conditioning) on a call for cooling,thats the cooling circuit. You can also close the fan switch Rc-G to operate the fan independently of the air conditioner. Newer systems just use R for both heat and cooling as all the controls are incorporated into one control board. The C(common) is just used to complete a circuit. This picture shows what happens at the furnace old systems used 2 different power sources.
You need to confirm that the new stat will accommodate Rh and Rc systems.  (https://i.stack.imgur.com/vhq0k.jpg)