Plot hole: It's stated that only objects or people infused with the chronoton serum can pass through the time barriers, which is why the equipment belts and hyposprays had to be specially modified (these wouldn't be needed if the chronotons were producing a protective field wider than the infused object). But some un-infused artificial objects, such as clothing and com-badges, can pass through them without vanishing. This is most obviously seen in the scene when Chakotay pulls the captain through the barrier, since he doesn't even wait long enough for the serum to spread throughout her entire body, yet alone to perspire into her uniform.
Plot hole: Tuvok says that he was born on Stardate 38774. But he was born in 2264, when Stardates were only counted in the thousands, not the tens of thousands. If he were really born on SD 38774, that would mean he was born only a few years before the Enterprise-D was launched.
Visible crew/equipment: Approximately nine minutes into this episode, as Tuvok approaches the crazed Hirogen, part of the body of a rather plump man wearing what appear to be Blue Jeans, a polo shirt, and white sneakers is visible on the very left of the screen.
Other mistake: In the 3D cinema scenes in the Holodeck, Tom Paris and B'Elanna Torres can be seen using blue/cyan anaglyph glasses and the movie screen can be seen to be in blue/cyan also. This is fine and to be expected. However the shot in which the projection room can be seen behind the couple show the projection is through magenta/green filters.
Continuity mistake: In season 6 episode 10 'Pathfinder', Admiral Owen Paris has the rank of Full Admiral. In season 7 episode 6 'Inside Man', the same Character is seen as being only a Vice Admiral. In the season 7 season episode 'Endgame', the last episode of the show, he is once again seen as being a Full Admiral.