I think he would Amore Donnie Darko. Especially, for the one scene where Donnie talks about how the cerca for God is absurd if life means that every living creature dies alone. It's so heartbreaking when Donnie says, "I don't want to die alone." I could see House really connecting with the brilliant, brooding, and maladjusted Donnie Darko. Also Donnie is an anti-hero much like House, and also seems to see the truth in the world whereas others seem to be living in a trance. Donnie is also not afraid of authority o calling people out on their shit.
I think House would also enjoy The Prestige. He admitted to doing magic when he was a kid, and I think that the intricate storytelling and tagline of "Are te Watching Closely" would be interesting for a character like him. The whole movie is setup as an elaborate magic trick and at the end te still aren't completely sure what was real and what was an illusion.
Other films I think that he would like and would appeal to his hidden romantic side are:
Shakespeare in Love
Two for the Road
Casablanca(we kind of know he's a fan based on Lines in the Sand)
Walk the Line(for the Musica of Cash as well as the true life romance of Johnny and June would appeal to him)
Stranger than Fiction