Yes and no. It will increase the resistance to the drivetrain and reduce maximum attainable speed. However if only the limited slip differential is bad then there is no real affect.
A bad rear differential causes vibration, noise, and fluid leaks. This can impact a car's acceleration and driveability, and cause very serious problems if not addressed.
If it is a Posi then the friction plates are going bad.
Check the oil level in the rear differential, drive shafts could also be bad.
You have more than one ABS wheel speed sensor. Probably three sensors, one for each front wheel and one for the rear differential. If the ABS light is on, have your local auto parts store read the code(s). The code(s) will help direct you to which speed sensor is bad if in fact you have a bad speed sensor.
The noise is due to bad gears or bearings in your rear differential.
Bad seal or loose drain plug.
it could but is not likely to the transfer case transfers power to the front diff, not the rear
I would suspect a defective universal joint on the driveshaft. Can also be a bad wheel bearing or axle bearing. And lastly, a problem with the rear differential. Check the gear lube level in the differential first.
A bearing that has gone bad. A universal joint that has worn out allows play in the differential gear.
Noises: growl, whine, squeak, moan, hum, droan, popping It could also shake, or have fluid leaking out.
The sensor in the differential may be bad. Also the wires going to the sensor may have rubbed against a brake line causing them to short. The sensor in mounted on top of the differential and reads the vehicle speed.
There are various seals in a differential assembly. Any one of these could be bad which isn't a big deal. Just needs to be replaced.