Maestro Sonata: Two quick points to add to your thoughts: it is résumé, not rèsumè, and the verb it comes from is actually résumer, not resumè.

porsche10x: I really have to disagree with using the word "soufflé" as an example to support your argument. That is another french word, and it has an accent solely because it has been borrowed directly from another language, just like "résumé," "café," or "naïveté." English uses almost no diacritical marks other than on loanwords from other languages, unlike some other languages that have more regular rules regarding which vowels get stressed on a word where the accent is required to show departure from the norm or to follow other obligate orthographical rules.

I agree that the accents should be all or nothing on résumé. If you are going to bother borrowing one from French, borrow both. Otherwise, leave it unaccented. As a French and English speaker, my opinion is that "resumé" looks misspelled in both languages. While your opinion may be that it does not, you may not know the background of the person to whom you are submitting a résumé. Lots of people in this world speak French, and using the single accent appears ignorant.

If you're going to put accents at all, put both (résumé), because this word is borrowed from French, and in French it has both. And, no, the first accent would not be grave. The difference between é, e and è is in the pronunciation, and they all sound different (think of the words "le," "théâtre," and "liège.") If you drop the accents, you're already anglicizing the word, so you might as well drop both. Or, keep both if you want to preserve the original French spelling. "Resumé" just looks like an incorrect hybrid.