Why do some English speakers pronounce “fête” as “fate”?

Warning: The following explanation is pure speculation: I am not a linguist, just a French with an amateur interest in linguistics.

In French, fête was written feste (or feſte) before the 18th century, and its /ɛ/ was¹ long to compensate for the lost s (see this wikipedia page). So Its pronunciation at time of its arrival in English was probably /fɛːt(ə)/.

Then, if one looks at Wikipdia's article on the Great Vowel Shift, one sees a transition in English from /ɛː/ (17th century) to /eː/ (18th century), then /eɪ/ (from 19th century). This would perfectly explain the modern (English) pronunciation of fête as /feɪt/.

By the way, as a native French, I'm really used to hear the native English speakers pronounce French /ɛ/ as /eɪ/, both in borrowed words and when they speak French. To me, it is part as the English accent as strongly as /ɹ/ for /ʁ/. So the short answer is maybe : because they say a French word with an English accent [insert Monty Python Holy Grail jokes here ;-)]

¹: Some French speakers still distinguish between /ɛː/ and /ɛ/, but it is no longer part of the standard French pronunciation.

Does this derive from a confusion between the acute and circumflex accents, or is there some other reason for this?

I doubt it's to do with confusion over the accents, rather not really caring about them. The pronunciation likely evolved from /fɛt/ "fet to /feɪt/ "fate" independently of dropping the accent. It seems unlikely we would decide to drop the accent and only then sitting down (with a nice cup of tea) to decide how that word should be pronounced.

It's interesting to see the word is fairly recent; Online Etymology Dictionary says the English word is from 1754, "Apparently first used in English by Horace Walpole (1717-1797)".

Thinking about the variants of e in French, we seem to pronounce é quite properly, e.g. café. But in my experience, I've never heard élite as eɪˈliːt. I don't know how widely this exception to the general rule exists, or whether there are others like it.

Similarly, I usually hear words like crème enunciated properly.

Taking common words where the e-circumflex is often mispronounced in my locality, for example, crêpe and fête, I can only presume that this is caused by a misunderstanding of how circumflex accents work.

Though not directly related, another common mispronunciation is manœuvre as məˈnuːvə, rather than məˈnːəv.

It's a hard topic to cover the breadth of, because some words have been imported from Old French via Latin, for instance, and may be spelt the French way but pronounced like a Latin word.