Does felt paper allow a house to breathe? I'm concerned about moisture passing through the interior walls and getting trapped in the framing. My main concern is mold growth behind the drywall and backerboard in a bathroom. The felt is attached directly to the wall studs because of the type of siding.
In my opinion, yes. Felt is lapped just like lap siding, but certainly isn't airtight. Felt has been around for probably a century and is still used in some capacity on basically every home built today. However, if you're worried about making the wall really breathe, go with Tyvek instead.
Both will allow it to breath. Tyvek allows it to breathe more. Most experts list #15 asphalt felt as having a permeance of 4 or 5 perms. The permeance of housewraps varies from brand to brand; Tuff Weather Wrap has a permeance of 6 perms, Barricade has a permeance of 9 perms, and Tyvek has a permeance of 58 perms. Of course, the ability of asphalt felt or housewrap to resist liquid water penetration is a whole other issue; one can't assume that low permeance wraps resist liquid water better than high permeance wraps.