Going down the list here are some of my findings so far:
The Day The Earth Stood Still (1951)
Takes place on Earth. Tell me you haven't seen this a million times: "A traveler in a saucer-shaped space craft goes down to an alien planet to extend an olive branch but is met with hostility. He escapes the custody of the aliens and disguises himself as one of them in an effort to learn more about them and eventually make it back to his ship." In the conclusion of the film, Klaatu mentions a type of interplanetary community, and when Earth is ready they could join it. This sounds a little like the United Federation of Planets and the prime directive.
The Thing (1951)
This movie also takes place on earth. However, besides the "Monster of the Day" plot, what is similar between this film and Trek is the organic chemistry between the characters. All of the characters in this film, both male and female, military and civilian, treat each other like equals. There is no "a woman's place is behind a stove"-style dialogue. Everyone's opinion is valued and all the characters (save for a few scientists) are treated as intelligent. Also, if you listen to the power generator start up, it's the phaser/torpedo firing sound.
When Worlds Collide (1951)
This film takes place mostly on earth. Not much that is Trek-like. However, the space ark has a tabletop-style control panel with a middle viewscreen that is attached to forward, side, and rear mounted cameras. And hey, John Hoyt!
Cat-Women of the Moon (1953)
Takes place on the moon. Not a very scientific film. It's like Trek in that a group of explorers stumble upon a dying civilization who wants to seduce them in order to steal their ship and revive their people. Also it features a female
navigator who has a hairstyle similar to Number One:
War Of The Worlds (1953)
Once again, takes place on Earth. Aliens come down and zap puny Earthlings. Star Trek did borrow a lot of sound effects from this movie. Also, if you take the Martian Heat Rays...
Add 1964 Bell Picturephones...
You get the Gooseneck Viewer...
And...
If you remove the scenes on Earth, the sexist jokes, and make the Monster scarier,
Fire Maidens of Outer Space (1956) is a half-way decent season 3 episode of Trek: