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.