Wrong proof in a research paper, will the paper be rejected?
First, notice that the question asked in the title has already been answered: your paper has not been rejected solely because an incorrect proof has been found.  So your real question seems to be "What do I do now?"  

The obvious answer seems to be to do what you have been advised: resubmit the paper with the faulty proof removed.  To me an "unproved theorem" sounds a bit self-contradictory; although there are some things that you might want to informally describe that way, using that language in a paper seems to be asking for trouble.  Rather you should decide whether you want to include the statement that you now realize you cannot prove in some form, e.g. as a question or a conjecture.  If you think the work would be much more valuable if the statement was proved, you might want to delay resubmission while you make a more concerted effort to prove it.  Or you may decide that without the theorem you had, the paper is not in your opinion worth publishing.  All this is up to you.