I have a collapsed arch in one of my feet due to a past injury. I have doctor-prescribed orthotics and have been cleared for any and all exercise activity. My primary goal is weight loss. It seems that most cardio machines cause me foot (and consequent leg) pain, however. What exercises can be done for weight loss that don't involve continuous pressure on my feet?
Strength training plus pool running
You won't get a definitive answer to your question, because there are so many options that could possibly work for you, and here's one!
Strength training (using a small set of full-body barbell exercises, like in Starting Strength, or Stronglifts 5x5) has been used by many people as an effective weight (fat) loss program. Lifting heavy uses a lot of calories, adds muscle, and removes fat. It also increases bone density. The movements are slow and controlled, so if you're in proper shoes with your orthotics, you should be able to hold your feet, ankles, and tibias in a less painful position. The ideal pressure distribution during a squat is actually back toward the heels and towards the outsides of your feet, which should help avoid pronation.
If you want to add conditioning work into the mix, pool running is a good option. There is no impact. I've used this during recovery from a tibial stress fracture. It's been used by olympic athletes to supplement or replace their land running during injuries while training for long distance running events.
I have a similar problem in that I have very low arches that are prone to injury from repeated stress. I have found that one of the absolute best exercises to utilize HIIT without foot strain is indoor rowing. the motion of rowing makes it ideal for intervals, and works the upperbody and core, as opposed to other low impact excercises like cycling. Another excellent low impact option is swimming, if you have access to facilities. switching between rowing, swimming and cycling should also help in preventing unnecessary foot strain. this link has several HIIT formats that work will with rowing.