The example 'don't stop believing' isn't using the present continuous tense. It is just using the gerund, which is very common in English. Erik gave you an example in his answer 'Don't stop studying English'. Again it is the gerund and not the present continuous being used there. The present continuous tense uses the present tense conjugation of 'to be' and the present participle, e.g. I am studying English at the moment. You can use 'to believe' in the present continuous tense .e.g He is believing in himself more and more' but other tenses are often more appropriate, which is what I think what you read was trying to say.

It depends on context, but you can use it that way. The usage in the Journey song is grammatically correct.