You're conflating the term 'virus.' Religion is mimetic, in that it behaves like a virus, but not everything that behaves like a virus is an illness. Some other examples of memes include fashion, catch phrases, and technologies.
There is no known inflation of the definition of "virus" in the preceding argument. Going by one of the many definitions of the word virus in the dictionary, Dawkins could have intended to mean that religion "copies itself and typically has a detrimental effect", as the dictionary allows us to use.
"Mental Illness" is not a well-defined term, but in its broadest sense it is a disorder of the brain that affects mood and behaviour in a harmful way. Religion may be a part of this, but if conscious beliefs can be classified as mental illness, the term becomes so broad as to be meaningless.