Body tape for weather?

They don't make a tape for that purpose, however around me (rust belt state) the popular choice is to clean up the rust, apply converter and then wipe the area with alcohol or wax and grease remover then carefully apply aluminum foil tape. This isn't common duct tape but thin aluminum with an adhesive backing. Commonly use to seal heating ducts and vents. The most popular stuff is 322 multi-purpose version by Nashua. If applied and burnished in it seems to last a couple years. However most rust on vehicles starts from the inside of the panel and by the time you see a bubble or hole the rust is very much larger than that spot. What you might be able to do to slow that down would be to get one of the kits from Fluid Film, Krown, NH OIL or similar, use that to spray into the area and the back of the panel. That would seal the rust away from the air and slow it down a lot. Then apply the tape after cleaning the area. I've seen all sorts of these repairs, some with the tape just torn off and stuck on to ones where they cut it into stars or to look like bullet holes. Very common on winter rats around my area. Some even buy paint and spray the tape to hide it better.

The rust has already gone through bottom of door, but hadn't gotten to inner layer of metal to the inside. I want to prevent that. I would get door repaired but can't afford to not have my transportation for the better part of a week. I just want to get a few more years out of car if the repair costs remain fair.

What does grease remover do? Not degreaser? I am not sure about a metal tape. I don't want to scratch anything.

The tape won't scratch anything, and even if it does, rust repair is going to require repainting most of the panel so any scratches would be gone. Plus if the door is rusted through they won't repair it, they will simply find a used one and install it then paint that door and blend the color into the body. That is common procedure with rust, it's faster and much less money overall than cutting out the rust and welding in new metal. Especially on a car that old. Doing patch panel repair or installing a door skin would be about 2-3 times the cost of installing and painting the used door. If you are handy with tools you could even DIY replacing the door. Also for that type of repair you can easily exceed the ACV of the vehicle as it gets expensive very fast. I would suggest you go on car-part.com and enter the information for the vehicle, I like to use states where they don't use salt or brine on the roads. See if you can find a replacement door, and have it shipped, if you look around a few states you might even find one that is the same (or very close) color. Then swap it over and if the color doesn't bother you, drive it. If you plan on having it painted, set down before you talk to the shop. Paint is very expensive.

I already got a $700 quote to repair the door without replacing it. It's way out of town and I still need to get to work in the mean time. I just want to stop rust and protect from moisure and salt as long as possible.

WOW, that is a great price for rust. Here anything over a dime sized spot will cost $1,000 minimum. Did they explain what they would do? The repairs I do are generally cutting out all the rotted steel, welding in a patch and then the rest of the work. Or if it's an available part, I'll remove the rotted one and replace it. Rust is very hard to stop. That's why I try to keep the vehicles we have filmed or waxed. Even then 10-12 years is about all they last without rotting away. When you see a 20 year old vehicle it's either a garage queen or bought out of state.

Yeah, he said he'd remove rusted areas and weld a metal piece in there and refinish outside type thing.

I would ask him for a time and rent or borrow a car, that is a fantastic price if he does that work. Plus the sooner it is done the less material has to be replaced. For that price around this area it would be them scraping out the rust, spraying a zinc primer inside, then filling the area with spray foam and a bit of bondo, then painting with a spray can...

I don't get a good sense from my mechanic how many more years my car may have so I don't know what's worth it. I try to keep annual car expenses under $1K.

That is always the big question on vehicles. Is it worth it to repair it? Normally I tell people that if the car is paid for, has a good body with no rot and hasn't been a money pit, that it's usually cheaper than payments. Then again I have a 2014 Journey that I've rebuilt the suspension, all new brakes, new engine and transmission in the past 2 years. That has 147K on it. It will be getting some body repair as well, but it's a spare vehicle at the moment.
The ultimate decision will be yours though. If it's running OK then the tape might do the trick for a while. It is possible to get decorative with it as well. Like the rust is at the bottom of the door, run a strip from the front fender to the rear quarter like a racing stripe. Then do the same on the other. Another option would be to install some of the stick on trim they sell for vehicles. Use something like the 3 inch black and just run that along the fenders and doors to go over the rust, the stuff is available at many parts stores and is usually about 30 dollars or so for a 30 ish foot roll.

I bought some aluminum tape at Walmart. It seems pretty thin. Like a moderate wind could punch through it. Is it supposed to be put on in layers?

It is thin so it conforms. However if you place the first piece so it is on the body then overlap the next piece half way you end up with a double layer that helps keep it in shape. It will actually take quite a bit to damage it, but not as much as the steel it is on.