The question you need to answer is what purpose the accent serves. Traditionally, in English and related languages, the accents have three purposes: to change pronunciation (for instance, the cedilla in the word soupçon causes the c to be pronounced like an s, not a k), to distinguish between homophones (as in French, between a, which means "have", and à, which means "at"), and to mark a change in historical spelling (the circumflex often indicates an s that has been lost to history, as in the French word hôpital, which used to be hospital).
My answer would be that an accent that shows a change in pronunciation (especially acute accents and diaereses) should be retained in formal writing, because we cannot know who will read it, and the reader may need these pronunciation aids; the same argument could be made for homophone distinctions, particularly in poetry.
However, if the accent marks a spelling change, it would depend on when the change occurred: there is no reason to continue to mark a spelling change that occurred before the word entered the English language. If the accent exists, for instance, on the French word, like contrôle, and it shows a letter was omitted long before the word was borrowed into English, then there is no reason to continue to mark the missing letter that was never part of the English spelling. There is, for instance, no mention on Oxford Dictionaries on-line of the spelling hôtel in English. The older word hostel came into the language in the middle ages, and is still spelled thus. The newer form hotel arrived in the mid-seventeenth century, when its French form was already spelled with a circumflex, and the two words were used differently. Hence, it is pointless to write hôtel; these are two separate words, not a change to an older form in English.
On the other hard, there are some newer conventions that are equally acceptable. The older spelling coöperate had already been completely lost by the time I learned to spell, twenty-five years ago, having been replaced by co-operate, which seems to do the same job, and so is perfectly reasonable. These days, cooperate seems to be quite common, and of course, there are no language police to say that it ought not be so; however, in formal writing, I still use the hyphen (as in e-mail; remember when there was still a hyphen there? And that was an originally English word! **And yes, it is okay to begin a sentence with the word "and"; your English teacher only said no because all young school children do it far to much).