I am going to go ahead and blur the line between Easter Eggs, tributes, impressions, and imitation. I will also add some obvious references just for the sake of discussion.
Here we go:
McCoy asks Nurse Chapel (off screen) to retrieve Kirk's vaccine (listen carefully- she may be the blonde woman Kirk oogles)
A brief eery tribute to the theme and ship sound effects during the widescreen establishing shot after Kirk meets McCoy
(Nearly) an original series communicator on board the Kelvin
Most of Spock's childhood/conversation with his father comes straight from "Yesteryear" (TAS).
Obv. some familiar lines:
Prime Spock says fascinating
Young Spock says fascinating
Scotty says he's giving it all its got
McCoy's I'm a doctor not a...
McCoy's your outta your vulcan mind line
McCoy's Greenblooded hobgoblin line
The way Kirk says "Bones" in the last three minues of the movie is ABSOLUTELY delivered with the Shatner swagger, head shake, and delivery. Chilling moment.
Uhura is assigned to the Farragut, which was the first ship Kirk serves on in TOS.
Kirk's taunting of Spock to the point of an emotional violent response plays out very similar to the episode This Side of Paradise (TOS).
Although never mentioned in any shows or movies, the writer's guide that served as the Star Trek bible, to help with continuity and technology, etc. stated that McCoy was divorced and had a daughter who was studying to be a nurse. This daughter is mentioned is in TAS episode survivor. This movie marks the first live action mention of his divorce.
Uhura's first name, Nyota has been uttered for the first time in all of Trek. It has been adopted into canon yet the source was Roddenburry and Nichelle Nichols who came up with the name when developing a backstory. Nyota means "star" in Swahili.
Sulu's fencing was an obvious reference to Naked Time (TOS).
The plot device of McCoy giving Kirk a hypospray to "give him the symptoms" of various diseases comes up frequently in TOS, the most notable being Amok Time (Makes Kirk appear to be dead).
McCoy's seeming fear of flying seems to be a reference to his TOS fear of transporters and is at least related to his disdain for technology.
I am sure there are more!