Heil furnace fan just comes on by itself. Never happened before. However I replace the thermostat in February and was wonder if that may be the issue.

Is this by chance an oil burner? I have seen on some oil burners where you cannot use a power robbing thermostat, as it will mess with the fan circuit control, causing the blower to kick on and off for no good reason. Only fix is to use a non power robbing thermostat. This is a long shot, but something to keep in mind. If you have the install book for the furnace, it would tell about the stat if it were an issue.

No its LP