The ac compressor runs when the furnace is in heat mode new thermostat installed

Is it a gas heater or a heat pump? If it is a gas heater, the wiring is wrong. If it is a heat pump, the compressor should be running.

You have wired it incorrectly. We need to know the colors of each wire and where they were attached tio the old thermostat and which new stat you bought. I suspect your blue wire is the problem.