Why is hot air not shutting off when everything is off?

SOURCE: car brake lights won t turn off when car is shut down 1996 Honda Odyssey

It should be your brake light switch. There is a little plate welded to the arm of the pedal that has a hole with a rubber stopper. The rubber stopper is most likely missing, or damaged. You should be able to take a flashlight and see the plate and the switch. If the hole is plugged up, you will need the brake light switch, if not, replace the plug.

SOURCE: Front end noise - Basically really loud "howl" when driving

If its a howl it is the wheel bearings

SOURCE: 1999 honda odyssey won't start, has power but won't start.

It's the neutral safety switch. Have your mechanic adjust it, or replace it if it's bad. For now try starting it in neutral instead of park or even try pushing the shifter up all the way (in park) while you turn the key.

SOURCE: Can't find the ''cabin air filter'' on my 1996 Honda Odyssey

A 1996 Honda Odyssey is not equipped with a cabin air filter. If I was looking for a restriction in the cabin air, the first place I would look is on the inlet side of the A/C condensor. ALL cabin air travels through the A/C condensor ALL of the time. (The heater core only sees air when the temp door is not directing the air around it and always after it has been through the condensor first.) I have seen gunk as hard as concrete clogging the condensor just from layers of dirt that collects on the wet coils then dries.

Testimonial: "Dear dttech, Thanks for the tip on the possible blockage on the inlet side of the AC condensor. Makes alot of sense and I will check there tonight. Is is difficult to access that area and how can I clean it once I get to it. Do I have to completely remove the condensor ? Thanks again for your help."

SOURCE: 1999 Honda Accord heater not blowing hot air

Sounds like a thermostat or cooling system issue. Change the fluid and the thermostat. It will do your engine good so it's money well spent.