So I changed my front brake pads and had problems pushing in the passenger side piston. I eventually got it and everything on it started shaking. The passenger side was super hot, so I figured I had a warped rotor. I changed that and it stopped for a couple days. It did it again, so I changed the caliper and it again stopped for a couple days. It started back up and I noticed the caliper would press but wouldn't retract. I was thinking it was defective, so I exchanged it and it is still happening. Any idea why it isn't behaving correctly?
The heat is happening because the caliper is not retracting properly. Normally, I'd suggest replacing the caliper - they're cheap on most (not high performance) cars.
However, you state that the you replaced the caliper.
Did you verify that the piston(s) in the new caliper move freely? If you're buying aftermarket calipers, there's a decent chance that you got a bad one. It's worth verifying. Because, honestly, a stuck caliper is, far and away, the most likely cause of this.
The next option would be your brake master cylinder. However, since the other three brakes are, I assume, functioning correctly, that should not be the problem.
The ABS controller technically could be keeping the brake down on that wheel if it has an internal failure. I have never witnessed this, or even heard of it happening, but it is technically possible.
Are you sure you have the correct rotor - not too thick for this car - and pads? If the pads and rotor aren't matched to the caliper, they could just be too wide, which would cause the same problem.
Check the slide pins - if this is a sliding caliper - and verify that they move smoothly. If they're rough or hard to move, they could cause the caliper to stick.
Pull the caliper, disconnect the brake line and verify that the pistons move.
Check the part numbers for the rotor and pads, as printed / stamped into the pads and rotor - and verify that they are correct for your vehicle.
But, now that I'm thinking about it - the slide pins are a likely culprit, and they're not necessarily replaced when you replace the caliper. They should move very, very easily. Start there.