When Ted is telling Barney that Ashlee is married, Barney is hanging out with a girl. Right before Barney says "Open your brain-tank, bro...", she puts her hand on his left shoulder. When the camera angle changes behind Barney, not only her hand is not there anymore, she is not there anymore. When the angle changes again, she is back. A shot later, her hand jumps to his right shoulder.
In the real world, the fact that the floor in the apartment was crooked would have to be disclosed before the sale could take place.
The first time Marshall, Lily and Robin enter the tilted apartment, it's clearly tilted down from right to left. When Marshall hangs a picture "level" (parallel to the tilted floor), it should therefore shift to the left, but instead it shifts "uphill" to the right. In all subsequent scenes the apartment is "correctly" tilted down from left to right.
While talking about how the floor of the new apartment is crooked, the three friends are drinking champagne but the liquid in the glasses does not tilt to level itself like any liquid would naturally do when placed in a glass on a slope.
When Marshall plays Ted the messages and the 'real' version of the previous nights events emerge, Ted is drunk when entering the club. He had met Barney there without having had anything to drink
Ted and Barney ask the bouncer if they can get into the club. He says he would if it was just them. But, because they have dates he cannot because there are too many women in the club already. Seconds after this, the bouncer lets two women into the club.
Ashlee pinches Ted, even though the illustration on his t-shirt has large portions of green. To be fair, she may not have seen the t-shirt from where she was standing, behind Ted.