I have Heil central air system. My blower motor (fan) is been running continuously, either in AC/HEAT OR ON/AUTO MODE. Only way to shut the fan is by switching the emergency switch to off.

Does the burner come on when in call for heat?

Yes, the burner does come on when called for heat...

It could be the fan and limit control or the A/C blower relay.

I looked around the blower motor and I don't see any relay. On the blower motor there is a square box with wires going to it and on top of this box there is a silver oval shape component, don't know if that's a relay or a capacitor. When I place the switch to on and auto on my thermostate, I hear a clicking sound from this oval shape component... any ideas what this thing is for? There is also engraving on it: 240v and some letters and numbers like uf5 and so on... My unit was installed in 1984, made by HEIL...

The silver oval thing is the capacitor.

Look in that box and see if there is a relay in there. It is possible the contacts are stuck/burnt together. Turn the power off before you go playing in the box!

Do you see a squar or rectangular shaped flat piece mounted on the inside of the top part of the furnace directly above the burners. There would be two wires connected to it.

I looked around the main junction box next to the red toggle sw and I saw the limit control box square shape with honeywell label on it... it has four wires going to it from the main junction box colored yellow-brown and black-black... while the fan blower motor was running in auto mode continuously, I tapped the limit control box and the fan stopped running right away... I'm thinking the limit control should be my fix... I also saw the wire diagram next to the main junction box and I did not see any relay on this diagram, just a capacitor on the blower motor and a transformer on the main junction and also thermostate for gas valve mounted next to the transformer and then the fan/limit control box... I will replace the limit control and see what heppens... thanks for helping me troubleshoot this problem...

I believe you are headed in the correct direction.

You have found your problem, glad we could help.