Holden is obsessed with the idea of authenticity, and he seems to feel that when people find out what religion a person belongs to, it makes them look at that person in a certain way.  After that, they are no longer able to see and appreciate the person for whom they really are.  In his own experience, Holden says that people are always trying to find out if he is Catholic, because his "last name is Irish, and most people of Irish descent are Catholics".  He cites the example of a time when he met a Catholic boy, Louis Shaney, at the Wooton School, with whom he

"struck up this conversation about tennis.  (Louis) was quite interested in tennis, and so was I.  He told me he went to the Nationals at Forest Hills every summer, and I told him I did too, and then we talked about certain hot-shot tennis players for quite a while.  He knew quite a lot about tennis, for a kid his age.  He really did.  Than, after a while, right in the middle of the goddam conversation, he asked me, 'Did you happen to notice where the Catholic church is in town, by any chance?'  The thing was, you could tell by the way he asked me that he was trying to find out if I was a Catholic...He was enjoying the conversation about tennis and all, but you could tell he would've enjoyed it more if I was a Catholic and all..."

Holden feels that once people have classified you as being of a certain religion, it changes the way they relate to you; their ability to see you as you really are is clouded by the attributes they begin to ascribe to you because of the group with which you are affiliated.  Holden actually says that "that kind of stuff" doesn't exactly "spoil" a conversation, but it makes it different.  He says that it "(doesn't ruin a) conversation or anything...but it sure as hell (doesn't) do it any good" (Chapter 15).

The quote you are refering to comes in Chapter Fifteen, when Holden meets the nuns and has a pleasant conversation with them. In a revealing flashback, he recalls having a great conversation with another boy called Louis Shaney who was at the same school as he was. they were having a really interesting conversation about tennis, and then he asks Holden if he knew where the Catholic church was, obviously trying to work out if Holden was a Catholic. Holden concludes that he would have enjoyed his conversation with Holden more if Holden was a Catholic. Note what Holden says about this:

That kind of stuff drives me crazy. I'm not saying it ruined our conversation or anything--it didn't--but it sure as hell didn't do it any good. that's why I was glad those two nuns didn't ask me if I was a Catholic. It wouldn't have spiled the conversation if they had, but it wouldn've been different, probably.

Holden thus shows how religion, and in this case, Catholicism, creates barriers. Let us remember that Holden is on a mission to try and form personal relations that are not "phony." Therefore we can understand his resentment towards anything that prevents such connections being established.

Holden Caufield and his author J. D. Salinger both had parents of different religious faiths, so they are sensitive to religious phoniness and stereotyping.  Salinger had one parent who was Catholic and one who was Jewish, so there was some religious identity issues and confusion.  The same it is with Holden: he says his father was a Catholic once, but now he is an atheist.

In Chapter 15, however, Holden loves the fact that the nuns (who outwardly show their Catholic faith by wearing the habit) do not ask him if he's Catholic.  He says:

That's why I was glad those two nuns didn't ask me if I was a Catholic. It wouldn't have spoiled the conversation if they had, but it would've been different, probably. I'm not saying I blame Catholics. I don't. I'd be the same way, probably, if I was a Catholic. It's just like those suitcases I was telling you about, in a way. All I'm saying is that it's no good for a nice conversation. That's all I'm saying.

Instead, the nuns discuss Mercutio, who--like Holden--refused to pick sides between the warring families.  Rather, Mercutio cursed both houses.  The same it is for Holden: he curses groups (religious or otherwise) who cluster together and have a mass identity.  Notice that Holden doesn't have to lie to the nuns; he engages them honestly because they're not phony.  He even gives them money.

Holden is an individual who aggressively asserts his own religious free agency and refuses to commit to the mass phoniness of the watered-down religion, prep-school "good old boy" network, or the materialistic consumer culture of the 1950s.  Instead, he enjoys conversations about literature with other book lovers who are not consumed with appearances, money, or status.