Tuberculosis is a fickle thing: people Satine was exposed to over a long period of time, like Zidler and the other girls, are more likely to have received it than Christian, in spite of intimacy. He may have had latent TB, alternatively.
Satine is diagnosed with TB. At the time, it was notorious for being contagious and incurable other than with rest, such as in a Sanatorium. No respecting business owner, nightclub or not, would encourage her to continue working with his other employees, let alone a duke and himself. Factor in all the patrons of the Moulin Rouge (many of whom Satine would have kissed or been in some way intimate with) and you have a Typhoid Mary situation on your hands.
The film is probably inspired by The Lady of the Camellias, which is also about a courtesan with TB, who falls in love with the narrator, and doesn't infect anyone. Plus, Moulin Rouge! is not at all realistic; aside from people randomly breaking into song (and singing stuff that wasn't yet written in 1900), it's quite often almost cartoonish. It should not be surprising that it portrays TB as non-infectious.
Who says Christian isn't infected? Tuberculosis can lie dormant for many years after infection. He may be heading for an early grave himself, as if the ending of the film needed to be sadder.