Which of the followings is correct?
There are still a couple of months left until the end of semester.
There is still a couple of months left until the end of semester.
There is/ are still a couple of months left until the end of semester
"Couple" is a number-transparent noun, meaning that the number of the whole noun phrase is determined not by the head of the NP, but by the number of the oblique (i.e. the complement of of).
Some number-transparent nouns can take singular and plural obliques (A lot of errors were made ~ "a lot of work was done").
But because "couple" means "two" or "a small number", it takes only plural obliques, so in your example, the correct verb is the plural "are".
In this case, couple is a collective noun, so the plural is correct.
There are still a couple of months left until the end of the semester.
Here are a few items for you to look over.
However, it should be pointed out that in colloquial/spoken English, it's extremely common to hear something like "There's a couple of beers left in the fridge if you want one," where there are is (incorrectly) contracted into there's. This might give the impression that "There is a couple of beers..." is correct, but it's not.
For example...in your example, you'd probably hear "There's a couple months left until the end of the semester" instead of "There are..." nine times out of ten, but formally it's incorrect.
The former sentence is grammatical, whereas the latter is not correct.
The phrase "a couple of" is used in informal English to refer to two or a small number of things or people. The phrase is followed by a plural noun as months in the sentence presented, so it takes a plural verb.