Kim Crosby and Robert Westenberg met while appearing in Into the Woods (playing Cinderella and her Prince respectively). While the characters broke up by the end of the play, the actors married and remain happily so.
RENT has a couple examples: Taye Diggs and Idina Menzel are the most well-known, having met during the play's original run in 1996 (they got married in 2003 and divorced ten years later). Also, Wilson Jermaine Heredia (Angel) dated his makeup artist while filming The Movie.
In the play's 2007 Australian run, Anthony Callea (Mark) and Tim Campbell (Roger) became a couple, adding a fun layer to the Ho Yay between the characters. They married in 2014.
William H. Macy and Felicity Huffman met during their stint with the Atlantic Theater Company and remain happily married to this day.
Stephen McGann played Dominic in Shamrocks and Crocodiles at the National Theatre, where he met playwright Heidi Thomas. Over twenty years later, she's head screenwriter on Call the Midwife, where he's playing Dr Turner. They're still happily married.
In the original Broadway production of A Little Night Music, ingenues Victoria Mallory (Anne) and Mark Lambert (Henrik), whose characters realize at the end that they love each other and then run away together, snuck off between shows one day to get married. Not only is the marriage still strong, their daughter Ramona played her mother's role in a major Broadway revival!
On the 25th anniversary tour of Les MisÚrables, barricade boys Jason Forbach (Feuilly) and Joe Spieldenner (Grantaire) began dating. When Forbach took over as Enjolras partway through the tour, showcasing the pair's chemistry, the delighted cries of Enjolras×Grantaire shippers could be heard 'round the world. They're now engaged.
Stephanie J. Block (Elphaba) and Sebastian Arcelus (Fiyero) met when they were doing Wicked. Their version of As Long As You're Mine is thus very believable.
Damian Lewis met his wife Helen McCrory while doing a production of Five Easy Pieces, where they played an adulterous couple. According to the director, they ended up sneaking off to jazz clubs during rehearsal, and the romance just continued. They're Happily Married with two children.
Sir Michael Redgrave and his wife, Rachel Kempson (later Lady Redgrave), fell in love while co-starring in Flowers of the Forest at the Liverpool Playhouse in 1935. In his autobiography, Redgrave recollects that the scene required him to extinguish a lamp which failed to turn off on cue due to an electrical malfunction, and then turned off unexpectedly when he and Rachel had just embraced on stage. The audience laughed a little, but once the performance was over, he and Rachel fell into each other's arms in earnest. The couple married soon after and remained more or less Happily Married until Redgrave's death.
Redgrave and Dame Edith Evans also had a brief affair during the production of As You Like It at the Old Vic in 1937, where he was playing Orlando and she, Rosalind. He was 28 (and married to Rachel) and she was 48. Redgrave later stated, "Edith always had a habit of falling in love with her leading men; with us it just went rather further." They broke up soon after production ended, but remained lifelong friends.