Better to ask this in the heating & airconditioniong. I have no idea what you mean by "the furnace is set to cool." Most systems use Red to White for Heat, Red to Yellow for cool, Red to Green to run the fan. Is this a new problem? Did someone screw with the thermostat wiring? Alas, when replacing thermostats it's always best to look at what the terminals on the old thermostat were marked as not everybody uses the same wire color as the terminal markings.

Thanks, Ron. You are right about the wiring. I have been switching the polarity at the various terminals after this problem originally popped up. I did find that polarity doesn't always prevent a 24v circuit from working normally. HVAC is def the place for this one. Tks again.

Check the wiring in the air handler as well. Open the panel and find the control board. There is an area on the control board similar to what is in the thermostat. For the thermostat wire, make sure Y is to Y, W is to W, G is to G and R is to R at both the thermostat and air handler. There will be another wire in the air handler that goes to your condenser outside, ensure that white wire lands on the yellow connection (not the white). Otherwise, the condenser outside will run during heating if it is on the white connection.