I recently updated to Xcode 8 and as I click on \"download\" option next to provisioning profile in Xcode/Preferences/Accounts/View Details menu it changes to gray and does not
Actually, all provided solutions could work or not, it depends on your luck, but mainly you have to keep delete account, clean up provision profiles like "mogile_oli"'s answer, and re-add them, but also, don't forget to reset signing identities:
Accounts -> Select Added Account -> Select Team and View Details (or double click team name) -> you will see Reset button if needed, click on it,
And you really need to try to build with Automatic Signing first then remove it, it seems can't be downloaded with button, for now just ignore the download button from this page, but better to use the dialogs while you try to build.
And always Quit Xcode and Re-open it after almost each action you do, After wasting like 2 days, I made it work like this !