Final solution at one place:

Important: The purpose of this post (and all the other posts by me) is to give a complete solution to all GMAT-Prep Verbal questions at one place. Sometimes students have to wade through dozens of posts to get to the final answer. My posts will give one complete and crisp solution required to arrive at the correct answer by eliminating the wrong one. Some of the content in these posts may have been taken from various other sources (discussion forums).

Radiocarbon dating of fossils taken from caves on islands along southeastern Alaska's coast suggest that at least a portion of the area was remaining ice-free during the last ice age.
(A) suggest that at least a portion of the area was remaining
(B) suggest at least a portion of the area remaining
(C) suggests that at least a portion of the area remained
(D) suggests at least a portion of the area that had remained
(E) suggests at least a portion of the area to have remained

The subject is 'dating (singular)', so the verb should be 'suggests (singular). This eliminates A and B.

With the word 'suggest', the word 'that' has to be used here as we are denoting a reporting verb (kind of) structure. This eliminates D and E.

In D, you can't use the past perfect ("had remained") unless there's some other, later past time marker (usually in the simple past tense) to which it refers. There's no second past thing here, so the past perfect is wrong.

Also, (D) and (E) change the meaning of the sentence. The radiocarbon dating suggests THE FACT THAT part of the area remained ice-free. It did NOT suggest the location of that area. If you say "suggests ... a portion of the area", then that would mean that the radiocarbon dating indicates the area itself.

C: correct