Thermoat heat and A/C blower work OK on "Auto", but can't run "fan on", I.e. Can't run fan continuously. Maybe I'm not wording the question properly, but I can't find this question anywhere on the 'net. Seems odd. Have a Lennox gas furnce. The green wire is connect at both the thermostat and furnace.

You say the green wire is hooked at stat and furnace, but is it hooked to the G terminal on both?

With some furnaces they will use 3 speeds on the blower-- high for cooling, medium for heating and low for continuous fan only. You will need to check for 24 volts at the furnace control board. With the thermostat set to fan on, you should have 24 volts on the G circuit. Check the voltage from C (common) to G. If you have voltage it's a board or blower wire that is bad.