At Carol and Susan's wedding, as Ross walks Carol in, you can hear the violinist accidentally brush the violin strings with the bow. The music itself was either played back from a tape or added in post production.
After Joey says "smell the fart acting," the cast messed this scene up several times because they kept laughing. But the editors didn't completely cut it out, so right after Joey says it, you can see Jennifer Aniston start to laugh before it goes to the next shot.
In the wedding reception, Susan asks Ross if he would like to dance and says that she will let him lead. When they stand up and begin to dance Susan is in the lead position (Left hand up) but when the camera angle changes Ross is in the lead position.
When Ross gets up to shake Rachel's mom's hand in Central Perk, the edge of the set is visible to the left of him where the crew can be seen watching a screen. [Only visible in the 16:9 widescreen version. The original 4:3 TV version was cropped to avoid these issues]