Deliberate mistake: When the Doctor begins to "fade" in the transporter room his mobile emitter fades with him. Since it's made of solid matter and is not a hologram, this shouldn't be possible.
Pathfinder - S6-E10
Deliberate mistake: When Barkley goes to see Admiral Paris at Starfleet Command, as Nicole tells the Admiral that Barkley won't go away, there's a photo of the Admiral's son, Tom, on the desk. Tom's Starfleet combadge is at his uniform's right side (it should be on his left). But during the conversation there's a closeup of this photo, and Tom's combadge is now at his left side. This is not a flipped shot. The combadge position in the photo was not edited in the 1st shot, though it was corrected in the 2nd shot. (00:24:00)
Basics (2) - S3-E1
Deliberate mistake: After Voyager's secondary phaser banks overload, everyone on the ship seems to be severely injured, but Seska's baby is perfectly fine. This is likely to be a conscious decision by the writers. (00:40:30)
Answer: If you include the original Star Trek series (1966) then there are several. The communicators used in the original series were before (and said to inspire) mobile phones. We currently do have teleportation technology but it currently only works on things the size of a few molecules. A "Cloaking device" also exists; it's a fabric that bends light through it, though it currently only works in infra-red. The Hypospray is real and was patented in 1960 - six years before the original series aired - it's actually called the Jet Injector. Faster Than Light travel is still a few decades off, but there are several real-world theories that look promising, including one that is remarkably similar to the method used in the Star Trek Universe called the Alcubeierre Drive that involves manipulating spacetime ahead and behind the ship and the ship "riding" it. Medical techniques and technologies have also advanced considerably; prosthetics particularity and we routinely have robots performing surgeries where absolute precision is needed. The "Shield" used in the series have a few primitive versions around. The Phasers used in the series are used but are not very powerful (nor will they ever be as powerful as the Star Trek version the laws of physics gets in the way) but rail-guns (using magnets to spin then propel a projectile) and particle accelerators like the Large Hadron Collider have been around for a while. The Replicator would require a nuclear fusion reactor and a nuclear fission reactor in something the size of a large oven and the Holo-deck wouldn't work at all based on our current understanding of physics so those are both still science fiction at the moment, but who knows!
Sanguis