The thread previously in the FAQ was littered with speculation and other troubleshooting so I've removed it and am posting just the information on how to check/fix/patch these two common issues.
Heater Control Valve Vacuum Switching Valve - HCV VSV
This little gem is mounted to the passenger side of the firewall (I assume that's correct for LHD vehicles as it is for RHD). The control assembly is mounted above and is easily recognized by the metal diaphragm can or bell which sits atop a VSV (see pictures below). Failure of the HCV itself is uncommon; Often the failure is the VSV. Diagnosis is simple:
1) Turn off the car and make sure the A/C is OFF.
2) Look at the HCV and not the location of the valve's control armature.
3) Turn the car on (or better yet have someone do it for you and check the armature's position again. If it hasn't moved and the engine is cool enough (don't burn yourself) you may try to move it manually... I recommend you do this with the engine OFF.
4) If the valve can be opened manually but does not open when the engine is running then you may try the VSV bypass described below.
Heater Control Valve and coolant routing:
HCV VSV bypass:
HCV VSV and Diaphragm (stock and bypassed):
Alternate mirrors of this information:
http://www.supras.com/06/techcenter/display.php?QID=67
http://www.ddmcse.com/turbo/heatervs.htm
http://www.toyota-supra.info/forums/...-problems.html
Heater Blower Resistor:
This is another common problem. There is a basic wire coil resistor cluster which sits within the blower duct behind the glove box; This is used to regulate blower speed and is cooled by the air moving through the duct. There is one coil for each speed: Low, Medium and High. These coils do tend to burn out over time. The replacement unit isn't very expensive (not last I checked anyway) and it's very easy to swap.
(Follow the first link below the image for complete instructions.)
http://www.rclambert.com/supra/blowe...%20removal.htm
http://www.cygnusx1.net/Supra/TechTips/blower_resistors.aspx
http://www.supras.com/06/techcenter/display.php?QID=56