What did they new Star Treks rip off of the original Star Wars movies exactly?
I should say "reference" rather than "rip off". What I mean is that it is clear, by JJ Abrams own admission no less, that he grew up a fan of Star Wars, but never watched the original Star Trek. So, when someone says that the trailer for Force Awakens looks no different than the Star Trek reboots, it's because they are all directed by Abrams. While he inserts characters names and places from Star Trek, his movies invoke far more memories of Star Wars than the source material he is supposed to be utilizing.
Here are similiarities that harken back to memories of Star Wars, and not Trek:
Leonard Nimoy's first appearance in the film as "Old Spock" coming out in a hooded robe to ward off a monster attacking Kirk lying on the ground is a literal recreation of Ben Kenobi coming out in a hooded robe to ward off the sand people from attacking Luke Skywalker lying on the ground.
Nero's ship blows up Vulcan, Death Star blows up Alderaan
The Federation officer uniforms from Into Darkness are carbon copies of Imperial officer uniforms
This one was pointed out by someone else:
Kirk and Khan propel towards a tiny shaft located on the admirals vessel which has a slim/none chance of success. This scene is very much like the thermal exhaust port of the death star which required pinpoint accuracy to succesfully destroy the space station. Ofcourse Kirks targetting computer in his helmet is destroyed halfway through which strongly ressembles Luke's decision to turn his off. The dialogue in this scene is almost identical in its flow to the one in star wars between rebel command and Luke(excluding any force related business)