Thermostats usually have the option of turning the fan on all the time, or just when the furnace is heating. On my thermostat it's labelled Fan: Auto, Fan: Cont. (continuous). (There may be a further option of hi speed when heating, low speed when not heating.) With today's digital thermostats you may have to search thru several menus to find the fan setting.
Or, the furnace may use up to 5 wires to the thermostat; if the thermostat cannot use all the wires, the furnace may consider an unconnected wire to be active. If the Fan-Continuous wire is unconnected it might cause this.
You will need to check your furnace schematic diagram or installation instructions to see what's needed, and compare it to the thermostat. Maybe something is miswired or loose.
Good luck with it.