Q:

Is religion still an important factor in the world of Fahrenheit 451?

There are several cases in Fahrenheit 451 where it is clear that Christianity, while evidently extremely rare, is still heard of. For instance, Faber notes that Jesus is still known, though in a much different form than before:

"It's as good as I remember. Lord, how they've changed it - in our 'parlours' these days. Christ is one of the 'family' now. I often wonder it God recognizes His own son the way we've dressed him up, or is it dressed him down? He's a regular peppermint stick now, all sugar-crystal and saccharine when he isn't making veiled references to certain commercial products that every worshipper absolutely needs."

At the same time, the Bible (a copy of which Montag saved from a house before it was burned) is evidently no longer read, and certainly not held in any high importance.  This is highlighted when Mildred asks Montag, "Who's more important, me or that Bible?"
Is religion at all a factor in the society, and if so, has it played a role in keeping old literary fables and stories alive, if only in the oral tradition? Has Christianity - or other religions - been a source of conflict and rebellion against the overwhelmingly anti-literary world?

A:

It isn't, at least not in the form we know it. The society presented in the book is one of leisure, easily consumed entertainment and carelessnes. Any thoughtprovoking material (i.e. books, especially critical or philosophical ones) has been banned, as it might lead to earnest and meaningful thoughts, discussions and questions. Of course the people fail to notice that this is how progress as a society is achieved, so they are stuck in an endless loop of meaningless entertainment programming. 
Religion doesn't fit in this hedonistic society at all, as faith and spirituality aren't 'fun' per se. Also religion and religious beliefs are tied to philosophical questions and considerations, so they would be regarded as equally evil as books by this society.
Of course, religion isn't banned or forbidden. It's rather instrumentalised by the entertainment industry, it's symbols and ideas turned into commercial byproducts and advertising media (as indicated by your quote). This way, even though religion still exists and people may use phrases such as 'my God', it doesn't hold any meaning any longer, therefore I make the case that a real religion in our understanding of the term doesn't exist any longer.
This is a common pattern in the book. Things that are considered a bad influence aren't banned or outlawed directly, they are rather turned around into a commercial product or they are disencouraged through peer-pressure and twisted rhetoric.
Of course, for those who have not yet succumbed to the world of leisure (such as the old man that Montag confides in), religion might still hold value and be a source of peace of mind or comfort. So religion still exists, but it doesn't have any meaning for the society at large. 

