How to dehydrate chili at home

Yes, they will ripen off the vine (and gain capsaician in the process) because they are a climacteric fruit. They do this best in a paper bag, like most peppers. However, there is a limit to this because they will eventually wilt.

I've dried out chillies a bunch of times, both with and without a fan-assisted oven. It's much easier with a fan oven, but not impossible with a conventional oven, but it is wasteful of energy in a conventional oven as you have to leave the door cracked. That and you must have an oven that has a very low setting.

The reason that heat works is that heating air reduces its relative humidity as warmer air can contain more moisture. Once the warmer air fills with moisture it will saturate, and won't be able to accept any more, and when that happens your chillies will stop drying. A fan oven will generally cycle the air out and bring in new air which will accept more moisture, but a conventional oven won't, so you either have to crack the door or open it every 10 minutes to bring fresh air in.

All in all it takes days to dry chillies in a fan oven with it being constantly on, and it will stink up your house while doing it. A more effective way (although I've never tried it) would be to use a desiccant like silica gel (available from hobby stores) in a sealed container to dry them out.

A good way to keep dried chillies dry in a moist environment is to put them in a sealed container with a bunch of rice grains in the bottom, the rice will absorb any moisture.