If, for example, you have a 720x576 video and make no edits at all but you change the resolution to 1280x720 or 1920x1080, will it cause a loss of quality? I know that it will probably increase the file size, but would it also affect the quality?
It could do. Most probably yes in a bad way. It may cause artifacts such as blotching and halos. Any defects in the original will be amplified. Random noise in a frame is almost certainly going to be spotted more easily.
There are fractal programs for enlarging stills but I'm not aware of anything currently on the market for video.
Yeah. All I ever hear or read about it is that it just won’t make the quality any better (obviously) and all that will happen is that the file size will be larger.
All my experience comes from having to prep stills of some of my stills images for giant advertising posters. (Sometimes more than fifteen to twenty feet high.) One thing to be aware of is the larger you go, the less contrast you should have as it has to be spread across a larger area. Even then the results of my work are seldom able to be seen at close quarters as the resulting posters get hung high above the ground. Often it's a close-up of a part of a frame I took maybe totaling 2MP from a 20MP or higher image. When I prepare these I correct them almost pixel by pixel before they get enlarged for printing and it can take some hours of going over an image before I'm happy with one. I don't think that would be viable for a video format although there may be professional software or hardware in the industry for automating just such a process, I've been out of the loop for too long to know. There are definitely automated black and white to colour converters. Studio Film Labs in London is one processing laboratory that has that equipment.
There are fractal programs for enlarging stills but I'm not aware of anything currently on the market for video
Not sure about the "fractal" part of it but have you seen or heard of Video Enhancer? I bought it some years ago (I see it's now free!!) and it does a pretty good job of enlarging the video frame size and increasing the resolution.