Why is "elite" pronounced /ɪˈliːt/ (rhyming with beet) and not /ɪˈlaɪt/ (rhyming with bite)? Most words that end in ite are pronounced with /aɪ/ — lite, trite, site, etc. — but elite is quite different. Why?
This has everything to do with the following:
- The language of origin of the word and
- The point at which the word entered the English language.
The reason that the words "light" and "might" and "site" have a diphthong is because they were present in spoken English during the Great Vowel Shift, which started in the 1400s and continued for a few hundred years. So words that had already entered the English language underwent that change in pronunciation. Those words could be of Anglo-Saxon, Norman French, or Latin origin. We see it across the board.
However, the word elite entered English towards the end of the period of the Great Vowel shift. The first attestation is in the mid 1700s (1738 in the OED). This is probably further complicated by the fact that French has long been the language of nobility, so there are certain pressures to maintain the pronunciation patterns of the origin language.
"Elite" joined the party a little bit late, so its pronunciation remained rather stable.
As Rjpond points out, elite is a modern borrowing. Words like crime, sublime, mime, etc. are older borrowings.
To note: There are early occurrences of the word "elite" in around 1400, with the meaning of a "bishop elect". This sense of the word died out and is now marked as archaic. So when "elite" was re-introduced (directly from French) in the 1700s, there was no existing pronunciation.
Small addendum: Already existing in English is another word that already occupies the same lexical space that elite would occupy, if the pronunciation were analogized to bite: alight, which dates back to Old English. (It's a lovely word.)
You might say that there is some linguistic pressure to avoid creating homophones: elite vs. alight.
"Elite" is a French borrow-word (élite). The 'i' is pronounced as in the French words égalité and fraternité.
English is a Germanic language but with many words derived from Latin. This is mainly credited to the Norman conquest of England which created two classes of people, one speaking a Germanic language and another speaking a Latin language. It is often noted that many of our words which are of Latin origin were historically used by upper classes - for example, the names of our livestock animals are all derived from Germanic (eg pig, cow) because lower classes farmed them, but the meat from these animals have names derived from Latin (pork, beef) because they were eaten by the upper classes. Words from this era tend to have been anglicised in pronunciation.
However, it still isn't surprising then that we have a French borrow-word to describe the upper class themselves - the elite. Although this is a more modern borrow-word, quite a lot of English words and expressions to describe classes are borrowed from classical languages, for example, aristocrat (French) and hoi polloi (Greek).