Unfortunately, there is no recommended safe, quick solution to drying a tent. Some people will shove them into tumble dryers but you should never do this as the heat can cause the material to rip or distort and it can even melt so just don’t do this.
Bringing the tent indoors or under shelter if the weather’s bad will help to speed the process up but you are still going to have to let nature take its course for the most part. A garage is an ideal space for you to dry your tent. Set up a makeshift clothes line in there and hang it out to dry. You can accelerate the process by running a dehumidifier in the garage but don’t place it close to a heater or use a hairdryer on it as this will cause the same problems as a tumble dryer.
If you don’t have a garage, then a spare room in your house will suffice. Once again, you could perhaps erect a makeshift clothes line or drape it over two chairs at either side of the room. Another suitable alternative would be to hang it over the shower curtain rail. It will be fine to leave a radiator on in the room as long as it’s not too close to the tent but any application of direct heat should once again be avoided.
If you are going to use a spare room in which to dry your tent, make sure that you put newspapers or some old towels or blankets underneath to catch the drips, otherwise your carpets or floor could get stained.
Other than that, it’s simply a question of time. If it’s a nylon tent, it shouldn’t actually take too long to dry. Canvas tents obviously take a bit longer.
The important thing is to be patient and wait until the tent is absolutely dry before you pack it away for the winter. If you don’t do this, mildew can form which will not only make for a horrible smell when you come to use your tent again but, worse still, it will damage your tent, ruin its waterproof coating and could cause it to leak. In spite of their being numerous products which help to minimise the effects of mildew, there is no cure for mildew damage.