Yes, it is possible to cure an infectious disease with another one. One example is the use of malaria to cure syphilis.
Before we knew about antibiotics, untreated syphilis could cause serious damage to the major organs and nervous system. However, it had been observed that a high fever could help cure it. Therefore Plasmodium vivax, the least aggressive form of malaria, was given to cause the patient to have a high fever for several days. The malaria was considered an okay risk, because of the availability of quinine treatment and the severity of syphilis would be fatal anyway.
The treatment won Julius Wagner-Jauregg a Nobel prize in medicine in 1927. Nowadays syphilis is much less of an issue, and can be treated with antibiotics.
Bethany has already said the main one I was thinking of. I cannot think of any others except in the case of maggots and gangrene. The maggots will eat the dead and rotting flesh that results from bacterial infection. It is unclear whether the maggots are able to produce toxins that kills the bacteria.