Is there any movies where the protagonist is an adult virgin?

No mention of The 40 Year Old Virgin?

Paul Schrader's Cat People (1982)

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The Name of the Rose, I think.

The Sessions (2012), which is actually hilarious. The trailer gives the correct impression.

Just about any old movie with a character that's not married....the old morality code.

The Nun's Story with Audrey Hepburn. Very entertaining. About a Belgian novitiate who goes to the Congo to work as a nurse. It probably has some racially problematic scenes. But it is an excellent movie.

The Island The clone of Ewan McGregor was a virgin (at least for most of the film).

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Did you know that in the 1980s movie PREDATOR the titular character was not originally portrayed by Kevin Peter Hall. It was in fact Jean-Claude Van Damme donning a much more insect inspired full body suit before he left the production which then led to the recasting and redesigning of the famous hunter.

That was mentioned right in the first post.

Doris Day movies, she's almost always a virgin even when married

Truman.
Star Wars.
Dumb & Dumber.

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Boldly going.

It's impossible to say conclusively unless it's stated explicitly, like in the movie 40-year-old virgin.

In the first Terminator (1984), Kyle Reese was explicitly stated to be a virgin then he impregnated Sarah Connor.

In Miyazaki's Nausicaa (1984), the antagonist Kushana says she is a virgin and since Nausicaa and Kushana are conceived by Miyazaki to be both sides of the same coin, I would believe that applies to her as well (Nausicaa is also stated to be an adult in the manga BTW). This also follows from the fact it was generally perceived as typical in Japan from 40 years ago for young adults to be virgins.

In general, in old movies set before the late 1960s, unmarried characters are virgins since it was typical in pre-1960s western culture to marry virgin.

My favorite is That Touch of Mink. In it we are supposed to believe Doris is a virgin trying to avoid sex with Cary Grant. Just the premise that Doris is a virgin here is hilarious in and of itself. She's a forty-year old woman at a beach resort with Cary Grant. What would she be waiting for?

Yup and it's 1962 at the start of the second sexual revolution and the birth control pill was becoming common place. I think Doris was just pretending

Bette Davis in The Virgin Queen (1955).

"Chauncey Gardener" in Being There (1979)?

In the one scene where Chauncey is provided a clear opportunity to have sex (as Shirley Maclaine throws herself at him), he can't seem to pick up on any of the cues... so we might assume that similar behavior would have applied to the rest of his life.
At one point he's also propositioned for gay sex, but fails to understand the offer.
Plus the idea (given at the end) that Chauncey may be some kind of pure, messianic being, might be interpreted that, as in the tradition of Christ, he lives out his life as a virgin.

Lawrence of Arabia, maybe? Correct me if I'm wrong, but T.E. Lawrence experts, biographers, etc. believe he was asexual and they have no solid evidence that he ever had any sexual encounters.

I don't know but maybe Julie & Julia,*2009
but the virgin is not the main character herself, but her idol, who wrote the book and in one of the scenes she said that she did not have an intimate relationship until she was 40 years old. The movie is great, by the way, if you are sad now and don't know how to move forward, definitely watch it

Kind of assuming both of these guys were virgins: Damien Karras - The Exorcist and Jesus - The Passion of the Christ

If he was asexual then I am not sure we can say "virgin" status applies.

That is because virtually no straight guy who never had sex with another man would say he is a "gay sex virgin" and virtually no gay man would say he is a "straight sex virgin." Since asexuals are not attracted to both kinds of sexual encounters then neither concept of virginity applies.