In the early 2000s, the NFL felt that touchdown celebrations had gotten out of control, delaying the game and promoting unsportsmanlike behavior. Some of the top "offenders" included Steve Smith (who "diapered and wiped" the football), Randy Moss (who mooned Packers fans at Lambeau) and Chad Johnson (who put on a hall of fame jacket and proposed to a cheerleader). So in 2006, the NFL outlawed "excessive celebration." Specifically, players were prohibited from any prolonged, excessive, or premeditated celebration, from using any type of prop while celebrating, from going to the ground during a celebration, and from engaging in a group celebration.
However, they clarified that the Lambeau Leap would still be allowed, as long as it was one player only that jumped.* Essentially, the Leap was grandfathered in as a celebration. The goal post dunk was also grandfathered in at the time, but in 2014 that was banned as well.
The 2006 celebration ban was not popular with the players or fans. And many players and fans were upset that the Packers were the only ones with a celebration that was still allowed, while their players were prohibited from celebrating.
This year (2017), the NFL has made a rule change again; this time, it has relaxed the excessive celebration rules. Players are now once again allowed to use the football as a prop, to celebrate on the ground, and to engage in group demonstrations. Offensive or prolonged celebrations, or demonstrations directed at an opponent, are still against the rules and will result in a penalty.
Before this year, it was debatable whether or not a Lambeau Leap-style celebration was within the rules at other stadiums. Players have attempted it in other stadiums, and it is usually not flagged, as long as it follows all the other rules (no props, no groups). With the new rule changes, a Lambeau Leap-style celebration should be allowed now in any stadium without question.
* Interesting side story: Several years ago I attended the annual "Packers Family Night" pre-pre-season scrimmage at Lambeau Field. Although it was strictly a practice scrimmage, there were referees there, treating it as if it was a real game (for their own practice). Early in the scrimmage, the offense scored, and the entire offense team did a Lambeau Leap. The refs, of course, threw the flag for excessive celebration.