We submitted a research paper to a journal for publication and we received the reports of the reviewers. However, the two reviewers pointed out a flaw in one of our proofs. Unfortunately, we cannot find another proof and we cannot also find an exemple in which the result doesn't work. One of the reviewers said that we can remove the result. The editor-in-chief asked us to send the revised version as new submission. What we can do in this case?
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.
You are worrying about a superficial consequence of doing mathematics research instead of what really matters, which is the research itself. This kind of thinking is an example of what's known as "putting the cart before the horse", and is in general a recipe for trouble.
What you should do is to do your best to try to prove the conjecture you thought you had already proved. If you succeed, great. If you fail, go back and revise the paper to reflect the new state of affairs. Treat this writing assignment as a completely new and independent project to the one you had before you discovered the flaw in the proof. Right now you seem to be emotionally attached to the idea of mentioning the "theorem" in your paper, but you should let go of that notion: you should decide on whether to mention it or not by taking a fresh look at the situation and asking yourself whether as a conjecture it really deserves to be mentioned, in the sense that it truly serves the interests of the readers and of the mathematical theory you are working in.
The bottom line is, don't second-guess the reviewers' advice. Just do the best job you can do to write the best paper you can, resubmit, and hope for the best.
What's the flaw in the proof? That can really matter. Have you looked at adding any extra assumptions which are required in order to make your proof work? If you can find such assumptions, you can include the proof of this weaker statement, and then have it as a conjecture that the result can be generalized (and give reasons/examples).
If you have no proof, anything could be the case. You could formulate a conjecture, this is legitimate. However, you have no idea whether it is true or not, and, as such, this is not a theorem. If you can show independence, you can take it (or its opposite) as axiom. However, hoping to getting an unproven "theorem" published is not serving science, and, by extension, not helping yourself, either.
If you consider the paper without this conjecture as too weak, you could collect more results before resubmitting - however, the editor recommends resubmission, so it does not sound as if all is lost.