I can't pick only three...so many tag teams over the years have not been given a real chance...or were just extremely under-rated, and never reached their potential (IMO)
The Hart Dynasty (DH Smith & Tyson Kidd) - Could have been the team to bring Tag Team Wrestling back from the dead. So many guys in the WWE are far less talented that DH Smith, yet he was released. They split up this team, had no idea how to use either guy. Smith was released, and Kidd hasn't done anything in the past 2 years. Complete waste.
Cryme Tyme (Shad & JTG) - People loved these guys. They were fun, they were different...and just when they started to get over, they were split up. Shad got a make-over and was almost immediately released. JTG hasn't done anything since. Another waste. Oh, and they never got the title shot that they were promised.
Lance Cade & Trevor Murdoch - Tons of potential. They had some great matches with the Hardys. Another team that was prematurely split up. Murdoch eventually got released, Cade was released and later rehired. Cade unfortunately passed away. But I wonder what these guys could have accomplished had they actually been given the opportunity.
The Dudebusters (Caylen Croft & Trent Barretta) - 2 young guys who could have definitely been the future of the tag team division. Croft was in development for YEARS and finally was brought up to the main roster, only to be released a few months later. They reminded me a lot of a young Edge & Christian. Too bad we never got to see them evolve. Trent Barretta is still hanging around...being wasted on Superstars and NXT.
Kronik (Brian Adams & Bryan Clarke) - During the Alliance, Kronik was brought in to feud with Undertaker and Kane. They had ONE match at a WWE pay per view and were immediately released. At the time, the match was said to be terrible, but looking back - it's not that serious. I'm sure if the WWE put some effort into developing their characters and hiding some of their flaws, they could have had a solid run.
Curt Hawkins & Zack Ryder - I don't remember how these guys were split up - but they were. Ryder is now on the verge of becoming a major player, while Hawkins is floundering. These guys could have really had a nice run with the tag titles for the past few years. Instead, Hawkins has teamed with like, 6 other people...and Ryder has just started to get over in the past 10 months or so.
The World's Greatest Tag Team (Haas & Benjamin) - Seriously, these guys were/are awesome. No reason they shouldn't still be in WWE today. 2 great workers in the ring, who could put on good matches with just about anyone.
Carlito & Primo - These guys were never really given much of a shot. These guys had a lot of potential, and I would much rather see Carlito with Primo, than Epico with Primo. Carlito has a great presence and a lot of charisma, and has the potential to be great in the ring. Primo is a great worker also, and his personality would have eventually come out while teaming with Carlito. Carlito was accused of being "lazy" and "unmotivated" - but when you've been with the company for as long as Carlito was, and were constantly being ignored and shoved to the background, you're probably be a bit disenchanted too.
The Basham Brothers (Doug Basham & Danny Basham) - Their pairing with Shaniqua wasn't exactly great...BUT they went on to be in JBL's "Cabinet". These 2 guys were good workers, and could have made a much bigger impact on the tag team division than they did in their careers. It's a shame they were never really given a shot.
Paul Burchill & William Regal - Burchill had a ton of potential, and Regal is hands down, one of, if not THE most under-rated guy in the industry. I really liked the pairing of these two. Regal was a great mentor, and Burchill really looked like a star at the time. They could have had a solid title run or two...but it never happened.
Doug Furnas & Phil LaFon - Not the most charasmatic guys - but they could go in the ring. I would have liked to see them and Owen Hart & the Bulldog REALLY go at it, and feud for the tag titles. They had a few matches, but they weren't as good as they could have been. I think the WWE shot themselves in the foot, but acknowledging Furnas & LaFon's lack of charisma on television constantly. They could have just been technical wrestlers like Bret, Benoit, etc. who the crowd respects for their ability in the ring. Instead, they were made to look like idiots.
Rhyno & Tajiri - When the WWE ran out of ideas for these guys, they had them team up for a while. I really liked this pairing. I usually don't like makeshift tag teams, but these guys had a history. They also complimented eachothers styles in the ring very well. Both guys were really over too back then. It's a shame nothing really ever became of this pairing.
The Mexicools (Super Crazy, Psicosis & Juventud Guerrera) - Minus the offensive "come to the ring on lawnmowers" gimmick they used - these guys were exciting to watch in the ring. At the time, they could have had some good feuds in the tag division. But they were basically just used a comedy jobbers - and eventually let go.
Chuck Palumbo & Sean O'Haire - Could have had a way better showing during the Invasion angle. O'Haire is one of the biggest wastes of talent the WWE has ever seen. Rumors always go around that he was extremely difficult to work with. If that's true, it's a shame - because the guy had a ton of potential. It sucks that his attitude might have really held back Chuck Palumbo from making a bigger impact during this time.
The APA (Faarooq & Bradshaw) - I think the APA are a very under-rated team. They were really overshadowed by the Dudleys, Hardys and Edge & Christian. But the APA and Too Cool were just as much a part of the tag team division at the time. They could have probably achieved more as a team.
Paul London & Brian Kendrick - Lots of potential, but for some reason or another, the WWE never really gave them a shot. Possibly their attitudes - who knows? Bottomline is, the fans were into them...and they never reached their potential.