I wouldn't think it would harm the dryer, I would be more concerned about the fumes harming the cat. Why not get a litter box,etc. and bring the baby inside?

The dryer will not be harmed, but Beagle_NC is possibly correct about gas fumes. The combustion products from the gas burner are pulled through the drum with the heated air and blown out the exhaust. However, I've heard that a properly adjusted gas burner doesn't generate a large amount of fumes and a dryer produces less than a gas range/oven. Don't know for sure, I've never had a gas dryer.

Seems like a costly way to get a little heat for the cat. Have you considered one of those electrically heated pet sleeping pads? They run at a very low (safe) heat, almost feels impercetible when you touch it, but make a good place for sleeping. Any big pet store should have one. They cost very little to run.

You might want to offer a small amount of high quality cat food (dry) to meet the extra caloric needs during harsh weather and to decrease the amount of energy needed to forage. Also, if you can set out very hot water once or twice a day, that will help. It will cool rapidly, but be drinkable while it does. Use plastic bowls so they don't break when frozen (even low plastic food storage or sandwich boxes will work just fine).

With all stray cats, try to stay safely separated from them (no contact) to reduce chance of a bite that might expose you to rabies. Of course, if you decide to adopt and bring the cat under care, then you will need to trap and take the cat to a vet for a health exam. After that petting and grooming and touching are just fine. Never take a new stray to the vet in your car unless it is confined in a secure pet carrier - while they may seem tame enough out of the car, once the engine starts that could change and you don't want to be trapped inside a moving vehicle with a desperate cat. I know this from experience!

L.

The dryer won't care but your gas bill surely will. This will be a very expensive way to provide comfort for your stray. Strongly advise considering another alternative.

On cold nights just put the cat in a carrier and bring it indoors and keep it in the carrier. You can let it go in the morning. When it gets down to 5F no animal should really be outside.
Some people thing that the cat's magical powers will protect it from the weather. But that just isn't true.

If you can't get kitty safely inside (for whatever reasons), do you have enclosed shelter of some kind? An open carrier or box or crate w/bedding?

You are kind to do this. Hope it works out for you and kitty.

You are better off using a ceramic heater with a adjustable thermostat with a timer. This will off set the costs of burning gas and putting extra wear on the dryer.

I wish there were more people like you out there!

You could put the stray in your bathroom overnight with a little bed, keeping him separate from your other cat.?

Also a child gate maybe? Then swap the cats around ? That way they can get used yo seeing each other and realise they're no threat to each other.

I did this with my dogs and a rescue rabbit now the 3 of them love each other! Good luck, just takes patience which you have.