RENT: (Tom) Collins is referred to as Collins by everyone, including his lover Angel. (This is an allusion to Colline, the character from La Bohème that he is an Expy of.)
In the musical I Can Get It For You Wholesale, Miss Marmelstein complains in her eponymous song about always being called by her last name. The list of names she'd prefer to be called includes her Embarrassing First Name, Yetta, and her middle name, "spelled T-E-S-S-Y-E."
This shows up a lot in The History Boys, which makes sense as it's set as the British school environment listed below. Lampshaded by Mrs. Lintott, who mentions that Hector's name is technically a nickname in its way. His real name is apparently Douglas, "but the only person I've heard call him that is his rather unexpected wife."
In Chess, Molokov (whose first name is Alexander or Ivan depending on the production) is referred to by his last name more often than not. The Broadway production's Playbill even refers to him as simply 'Molokov' while referring to the rest of the characters by first name.
Like in the source book, the revolutionaries in Les Misérables are referred to solely by their last names, with two exceptions: Marius Pontmercy (called Marius), and Jean Prouvaire (name never spoken aloud onstage and generally nicknamed Jehan in the novel).