3M makes an outdoor rated blue painters tape made of plastic. So it's pretty weatherproof, but still peels off easily. If you're looking for something to tape up all of the cracks from the outside, and removing the spring that might work. No guarantee that it will come off without damaging paint but it seems pretty likely as long as you don't peel it off while it's still too cold for that to work well.

But you'll still be losing a lot of heat through the glass, and having more layers with airspace between them will really help so putting a plastic shrink film over the whole inside will also help.

You could also think about replacing it with a single regular door with a side lite next to it. We replaced our 6ft sliding glass patio door with a 2.5 ft square window above a regular insulated wall section for 1/2, and then a 3 ft wide regular door, with a triple-pane window in it, for the other half. It looks great and the space inside of there is much warmer than it used to be.