What was the purpose of the "tail prop" on the Ilyushin Il-62 and why was it necessary? I read somewhere that the Ilyushin Il-62 had a "tail prop".  What was this device and why was it needed on this aircraft?
Are you asking about the rear support wheel? It is for supporting the aircraft and avoiding tilt on the ground.

Image from wikimedia commons. Autor:M Radzi Desa

As the aircraft had four engines in the aft, the cg would've been aft too; so they added a tail prop to prevent tilt on ground.

Interestingly, this doesn't seen to be unique for Il-62. Another aircraft with rear mounted engine, Sud Aviation Caravelle, has something similar to this. 

Image from wikimedia commons. Author: Eduard Marmet