I have three baseboard heaters that are controlled by a thermostat on the wall. The thermostat wire goes to a relay box that is attached to the electrical panel. The power wire for the system goes to the relay box from the breaker and then another wire is sent from the relay box to the heaters (don't know if they are wired in parallel or in series). The system will not work even though my field tester shows that both hot wires going to the baseboard heaters are live (220 volt heaters). When I switch the thermostat from program to manual, it still does not turn on the heaters. Is there some part within the relay box that could have gone bad and needs to be replaced?

You say both of the hot wires going to the heaters are live?
If the tester you are using is the type where you just hold it next to the wire
And it lights up... it may not be helpful.
A solenoid or a "wiggy" style tester or a multimeter would give better information.

I would bet the problem is with the relay. My guess is that the relay is a single pole relay
That makes and breaks one of the 220V feeds. The other 220V feed goes straight through to the heaters. You need to measure, with a wiggy or multimeter the voltage to ground of each hot wire on the "output" side of the relay. Each output wire should measure 120V to ground and 220V between them. If you do not measure 220V on the output wires it's time to replace the relay...
Get a better meter so you can be sure...

I got a heahy duty extionsion cord with a cirbreakr on one end to operate my portable 15100 BTU air conditioner . Can I use thi to operate an 1500 watt lifesmart infrared quartz heater?

Please provide more details, such as wire gauge of cord, and amp draw of AC unit.