Visible crew/equipment: When Mario is telling Luigi about getting a call from the café for the plumbing job at the start of the film, watch the cabinet with the glass doors behind Luigi. At one point, you can see the reflection of a crew member in a short-sleeve shirt for a few seconds.
Revealing mistake: When the meteorite strikes the Earth at the beginning of the film, there's an odd slip-up in the effect. About a half-second before it hits, you can sort of see it awkwardly shift slightly to the right for a split second. It's hard to catch, but once you see it, it's fairly noticeable.
Audio problem: Throughout the scene, where Mario and Luigi are driving to the plumbing job at the beginning of the film, there are multiple moments where their lips and dialogue don't match. It's primarily in the wideshots outside of their van.
Continuity mistake: When Koopa is pushed into the chair in the Devo Chamber, suddenly a puddle of goop appears on the floor, and you can see that Koopa is sitting atop an empty uniform that wasn't there before. (There was a short, deleted scene where Koopa used the chamber to reduce a worker to primordial slime, leaving the puddle on the ground and the empty uniform on the chair. However, as the scene was cut, the puddle and uniform appear out of nowhere in the finished film.)
Continuity mistake: When Toad is talking to Mario and Luigi in the jail cell, he has his arms dangling down through the grate into their cell. When the camera is looking up at Toad, his arms are sticking through grates that don't align with each other, but during overhead shots his arms are through grates in the same row.