Car: 2003 Mazda protege 5
For the past month, I’ve been hearing a sound that sounds similar to something knocking around and touching metal. I think its coming from my passenger right tire but there is no real way for me to tell. At first I thought the cause of the noise was because all my tires were near balding but the noise still persist even after new tires. The noise could be heard in drive and reverse mode while going 0-50mph. I never hear it on the highway, only on local roads. Attached is a sound clip of the noise I’m referring to. In the clip, the car starts off at a stop light then makes a right turn, then straight, stop, u-turn left, right turn again, and finally straight to end. I’ve taken it to a mechanic before and he told me it was because of my warp rims. My car doesn’t overly shake on local or highways so I don’t think that’s the case. Maybe my mechanic is not looking hard enough or I’m just overly paranoid and doubtful. Does anyone know what this sound means?
Sounds like exhaust pipe banging. Maybe a clip broken or broken motor mount.
I just tried to wiggle the exhaust to see if it’s loose but it seems pretty sturdy. If it’s the motor mount, wouldn’t my car be shaking heavily cause the engine would be rattling?
"all my tires were near balding"
If you haven’t done so already, GET NEW TIRES. Tires should NEVER be run to “near balding” condition. This is extremely dangerous to you and everyone around you.
“he told me it was because of my warp rims.”
“My car doesn’t overly shake on local or highways”
With respect, you have far more serious problems than the noise. Focus on the rims, the cause(s) of the shaking (I suspect you have more causes than just the rims), and worry about the noise last if it’s still there. This vehicle sounds unsafe to drive.
It may be something simple. Like a pebble is stuck somewhere inside the wheel. Some wheel designs are more prone to this than others. Sometimes a mechanic can remove the wheel and shake it around and will notice something rattling. Pebbles can get into the brake parts and rattle around too. So if the mechanic doesn’t find it in the wheel, they’ll check the brakes too, while the wheel is removed. If a pebble is the cause, with persistence they can usually find and remove it.
The same mountainbike: I got new tires. I replaced all four but the noise is still there. I honestly don’t feel that my rims are warped enough to be a problem. The car does not shake like its unbalanced. I can live with the noise, it doesn’t really bother me. My concern is not the noise though, it’s why is there a noise? Is something broken? Am I going to drive somewhere and a wheel is going to fly right off? A abnormal noise raises questions 
Georgesanjose: I’ll need to check if there is a pebble stuck along the breaks. So far I’ve tried changing each tire to my donut to find the culprit but I still hear it. That should rules out the wheels. I want to add, If the road is flat, I don’t hear the noise. I seem to only hear it if I’m going through bumps or making turns. If my trunk wasn’t empty, I’d suspect it was something rolling around inside my car 
Take it to a good chassis shop for an in-depth look-see. It’ll cost you an hours shop time, but then you’ll know exactly what you’re dealing with.
Meanwhile, don’t underestimate the effects of a bent rim. There’s a whole lot of inertial energy in a wheel/tire combination, and at high speeds that inertia if shaking or bouncing (you didn’t mention how it was bent) can really push & pull on the hub… thousands of times a minute. It can be enough to separate worn joints.
Take a wheel and tire not mounted on a car and try to shake it around with your hands. You’ll be surprised at how much force it takes.
I seem to only hear it if I'm going through bumps or making turns. If my trunk wasn't empty, I'd suspect it was something rolling around inside my car :P
Check the spare tire and jack too. One time I had a golf ball that rolled into a little nook by a rear wheel well, but inside the trunk, and the golf ball was rattling around like crazy going over bumps and turns. It took me a while to find it.