Notable that Triple H and Stephanie went through their entire on-screen relationship from start to finish before they began to date in real life. In one DVD interview, Triple H half-jokingly says that one reason for their happy marriage is that they got to do a dry run before the real thing.
(In)Famously, Chris Benoit and his wife, Nancy. They fell in love during an angle where they were having an affair behind the back of her then-husband, Kevin Sullivan - an angle Sullivan pitched.
Kayfabe being what it was back then, this angle involved Chris and Nancy being seen at the same clubs, eating at the same restaurants, and even staying in the same hotel rooms, appearing to all the world as though they were the genuine article. The fact that they hooked up for real probably made Kevin Sullivan the only wrestler to ever be shockingly swerved by his own angle. And given the... unpleasantness of a few years back, not a few conspiracy theorists armed with very-coincidental-but-still-interesting evidence have suggested that Sullivan may have been involved in the incident and staged it to look like a murder-suicide on Benoit's part. Booking his cheating wife and the man who stole her out of the picture, as it were.
Matt Hardy was Amy "Lita" Dumas's boyfriend both on-screen and off, but she cheated on him in Real Life with Adam "Edge" Copeland (who cheated on his wife, thus breaking up his second marriage in barely five years. The first was to Sean "Val Venis" Morley's sister Alannah. WWE had intended to make an angle out of that, but it was scrapped after Val reportedly had taken too long to come back from a neck injury). You know what happened...
Brock Lesnar (who had a fiancee and baby at home) met Sable during his 2003 run on SmackDown. Today, they are married.
John Morrison and Melina (on and off). They met during auditions for the third season of WWE Tough Enough (Morrison made it; Melina didn't but she eventually got signed anyway), and stayed together for over a decade before finally calling it a quits in 2015.
Billy Kidman and Torrie Wilson first got together in WCW after becoming a couple on-screen, then got married while they were in WWE, where they were never an on-screen couple (at their insistence). They have since divorced.
CM Punk dated Daffney Unger and later Traci Brooks during his time in Ring of Honor. Then, while in OVW, dated Maria Kanellis, but they split up due to WWE moving them to two separate brands (there are also rumors of him having an affair with Kelly Kelly around this time). Then, he briefly dated Lita. Then, he started a long term relationship with Beth Phoenix, which ended by 12/2011. He seems to have gotten back together with Lita due to them making a lot of public appearances together since Wrestlemania.
Punk has also reportedly had a fling with Mickie James and in 2013 started dating AJ Lee meaning he's dated AJ and her idol Lita in the same year. He has since married AJ.
Lita's ex Edge is dating Punk's ex, Beth Phoenix, later having a child with her. All these people date in circles.
The entire two-year feud between Tommy Dreamer and Raven in 1995-1997 was fueled by the simple premise of Raven hating Dreamer going back to when they were kids and Dreamer had been mean to Beulah McGillicutty when she was the fat girl at summer camp. Beulah debuted as a Heel at ECW Hostile City Showdown in April 1995. At House Party 96, January 5, announcer Joey Styles was trying to conduct an in-ring interview with Beulah and Raven's flunkies Stevie Richards and The Blue Meanie in which Beulah annnounced she was "pregnant." Raven proceeded to beat up Stevie and the Meanie until Beulah said, "It's Tommy's!," thus turning herself Face. Dreamer ran in and beat up Raven, including piledriving him on a STOP sign and a blueberry pie. Dreamer and Beulah would remain a couple on-screen and off until she left in Summer 1998 after The Dudley Boys broke her neck with a Dudley Death Drop. Dreamer, real name Thomas Laughlin, and Francine (Fournier), Beulah (Trisa Hayes)'s Arch-Enemy both on-screen and off, would have a relationship offscreen until Laughlin apparently changed his mind and went back to Hayes. Reportedly, Fournier was angry enough about this that Laughlin and Hayes had to have extra security at their wedding. This was what Joey was referring to when he said that Dreamer had "scorned" Francine, following Francine's interference in the Dreamer/Sandman-Dudley Boys main event at ECW One Night Stand, June 12, 2005.
Daniel Bryan and Brie Bella have been dating each other for years, and got married days after Wrestlemania XXX.
John Cena is currently dating the other Bella Twin (Nikki). He proposed to her in front of the entire audience at WrestleMania 33, she accepted, and they're currently engaged.
In his autobiographyA Lion's Tale: Around the World in Spandex, Chris Jericho claimed to have dated Kimona Wanalaya while in ECW in 1996 and two unidentified members of the Nitro Girls (a dance troupe who would entertain the crowds between matches and going into and coming out of commercial breaks on WCW Monday Nitro from 1997-1999), though none of these situations carried over into angles.
Rusev and Lana are engaged as well. This resulted in some nonsensical Real Life Writes the Plot moments in Kayfabe when WWE tripped over themselves thinking they had to respond to a TMZ report that the pair had announced their engagement (as in-universe Lana had betrayed Rusev to be with Dolph Ziggler at the time, while Summer Rae had paired off with Rusev, which got completely thrown out of the window when the TMZ story hit). You can read the bafflingly messed-up story here.
Ric Flair's daughter Charlotte was married to TNA star Bram, and was formerly married to an independent wrestler who worked with her brother Reid.
In a rare subversion for a wrestler/wrestler relationship, Natalya Neidhart and husband Tyson Kidd. In spite of working together in several different promotions before making it to the WWE together, wrestling ultimately wasn't the impetus for Nattie and TJ's relationship; they had known each other since childhood, and were together for well over a decade before they finally got married.
The nude photo scandal in February 2015 revealed that Seth Rollins was having an affair with NXT Diva Zahra Schreiber — while both were reportedly engaged (Seth's then-fiancee (and girlfriend of six years) was in fact partially responsible for said scandal). The only reason he wasn't punished was because of Triple H, though his new girlfriend did eventually get the shaft six months later, ironically for something that had occurred before she had even met Rollins. They continued dating afterwards for a bit before breaking up in February 2016 to little fanfare.
Paige had (short) relationships with numerous wrestlers on the roster while she was still down at FCW/WWE NXT, including Xavier Woods, Dean Ambrose, (possibly) Seth Rollins (which meant that Zahra Schreiber wasn't the first person he cheated with on his fiancee), and Brad Maddox (who would release some of Paige's sex tapes in mid-2017). Most infamously, she started dating Alberto Del Rio at the start of 2016, to great controversy. Del Rio was in the middle of a divorce and allegedly had a fling with Charlotte just a short while ago, but those weren't what disturbed everyone. It was the age difference between the two, and how the relationship coincided with changes in Paige's behavior, including getting two tattoos of the guy literally months into their relationship, two wellness policy violations, one after the other, and Paige deciding to propose to him not even a year into their relationship in another company's wrestling ring. The two continued to generate controversy for the next year (including the aforementioned sex tape scandal and a recording that caused an investigation into possible domestic abuse) before breaking off the engagement in October 2017.