I mounted some tires to use as a burners to do some sliding so I don't burn my dailyes the car is a 93 Miata which is a RWD. The tires I mounted on the back are both from a civic but the tire size is different, it is something like 185/70/14 while the ones that were on the back and are also currently on the front are 160/50/15 or something like that not quite sure. My question is as long as the tread depth, the wheel and tires are identical am I risking any damage to my differential?
No you should be ok with different sizes on the rear compared to the front.
Having each side the same is better for your diff. Especially for normal driving where you have traction on both wheels as the diff is constantly working to compensate for different rotational speeds.
However, doing burnouts and drifts is hard on your drive train anyway. If youre doing that with both wheels spinning then having different sizes would make little or no difference.
The most extreme workload for your diff is having one wheel with traction and the other spinning. In this case the tyre size would be irrelevant.
Having equal tyre sizes is most important on an all wheel drive car. Where it has front, rear and centre diffs.