So this argument could go on forever. Literally.

In reality, résumé, resumé, and resume are all correct. When companies (and magazines, etc.) brand themselves, they choose a dictionary to use and follow. When it comes to issues like this, they always defer to the first spelling listed in the dictionary. However, here's the catch: in the American Heritage Dictionary, it is resumé. In Webster's, it's résumé. There are discrepancies between every dictionary. So it all comes down to which dictionary and style you follow. Everyone is right.

If you work for a large company, chances are they have their own style guide that you should refer to, and when that can't answer your question they give you a style guide and dictionary to default on. Personally, I follow American Heritage because there's no reason for the first accent in résumé and there's no reason to pronounce the "e" at the end of resume if it doesn't have an accent. But that's just my style. So really, you're fine no matter what you use.