I removed the stuck cable from my 71 and attempted to lube it. It absolutely would not budge in its metal sheath until I put the bent end in the chuck of my variable speed drill, secured the other end in my vise, and slowly rocked it back and forth with the drill motor. After it twisted loose I removed it and greased it wtih lithium grease. No further problems except trying to get it set so that "Cold" is all the way off at the valve, and "Hot" is all the way open.
The heater control knobs are held on by small allen nuts. Just loosen them and the knobs pull straight off. But don't do this until you check out the linkage. For the temperature control, disconnect the linkage at the control valve on the firewall (engine side) and then attempt to move the control. If it works OK, then the control valve is stuck or sticky. If the linkage is still stiff, you can attempt to oil it or replace it. For the selector control, you will have to try the same ritual at the heater box. If you have very small hands and arms, you will be able to get to it with no problems. Normal people end up scratched and bleeding, cursing and swearing, while upside down with a gear lever up their back. No big deal, right?
Cables don't like grease. I would recommend a graphite-based lubricant such as a speedo cable lube (available in most auto parts stores). This stays slippery and does not get hard and gummy like grease. BUT, as previously mentioned, disconnect the cable from heater valve first, and actuate the lever, to determine whether the cable or the heater water valve is what is offering the resistance.. On my Spifire, the valve is stiff. Lubricating that with grease is what made the most improvement in my heater control's feel. BTW the Spit controls the amount of heat provided solely by its heater flow valve, not by an air mix door as on more modern designs. So, you shouldn't have to pull the heater box to repair a heater problem unless the core needs service. Cold drafts are usually caused by air leaks in the firewall grommets or tranny cover. A secondary problem could be bad or missing foam seals on the _direction_ control doors, check them while you have the dash apart.
I had mine jam up on my spit this past summer. A few weeks ago, I started investigating the problem, and found that the valve itself jammed. As I was gently forcing the cable, a metal cylinder(bushing?) fell out of the bottom of the valve. Can this thing be put back in, or is the valve toasted. I t doesn't seem to be leaking at all.