While Americans (and possibly others) pronounce this as "loo-tenant", folks from the UK pronounce it as "lef-tenant".
Etymonline indicates that spelling with lef- dates to the 14th century, but that the origins of that spelling (and presumably its associated pronunciation) are “mysterious”. The word comes originally from Old French, and according to the OED, Old French replaced word- and syllable-final [w] with [f]; for the Modern French word lieu, this is shown by an Old French spelling variant luef. Both forms, whyever they exist, just happened to stick.
'Lieutenant' comes from French lieu ('place') and tenant ('holding'). Some sources claim that 'lieutenant' had alternative spellings such as leftenant, leftenaunt, lieftenant, lieftenaunt etc., and that the ModE pronunciation with /f/ (BrE mostly) is a holdover from those spellings.
I thought the pronunciation with /f/ arose from the 'minim confusion'; in Middle English, both v and u were used interchangeably. According to Lexico, "the u at the end of Old French lieu was read and pronounced as a v, and the v later became an f". I can see how the v became [f] (see 'assimilation') so it sounds plausible to me. However, according to Etymonline, the OED rejects that theory.
The ModE pronunciation with /f/ means one of the following things:
The speakers of the French dialect lieutenant was borrowed from probably pronounced the ⟨u⟩ as [v] in some places and it took the devoicing from the following /t/ (cf. 'hafta' from have to)
Or, as orthographic ⟨u⟩ and ⟨v⟩ were often used interchangeably, Anglophones for some reason hypercorrected their pronunciation to match the orthography, so: /l(j)ɛu:ˈtenənt/ (or /l(j)ewˈtɛnənt/) → /l(j)evˈtɛnənt/ (hypercorrection) → Assimilation → /l(j)efˈtɛnənt/.
Or, the Anglophones confused the lieu with the English word leave (live) as and got the pronunciation /l(j)evˈtɛnənt/ instead of /l(j)uːˈtɛnənt/ and then later on the /v/ got devoiced to /f/
Or, the pronunciation with /f/ is a holdover from one of the spellings with an orthographic f
According to More Word Histories and Mysteries: From Aardvark to Zombie (American Heritage Dictionary), the origin of the pronunciation with /f/ “is not known with any certainty, but similar pronunciations are attested in Middle English times by such spellings as leuftenant, luffetenand, and levetenaunt”. Note the spelling with f and v.
The Old French word lieu had a rare variant form luef, and a form of Old French lieutenant using this rare form rather than lieu may have been picked up by Middle English speakers. In addition, the Old French pronunciation of the word lieu was something like (lyĕw), although this has developed into (lyœ) in Modern French. It is possible that Middle English speakers may have heard the final (w) of this word [...] as a (v) or (f) [...].
Both [v] (also [f]) and [w] are ‘labial sounds’—that is, made with the lips. So it's reasonable to say Middle English speakers confused both [v] and [w].
The quoted entry goes on to say that:
The Oxford English Dictionary suggests that use of the Middle English forms with f may also have been encouraged by an association of the first element [lieu] with other English words, such as the noun leave—a lieutenant being an officer who substitutes for another who is on leave or perhaps one who has the superior officer’s leave to take command when he is absent or otherwise unable to fulfill his functions.
As for the AmE pronunciation, John Algeo in The Origins and Development of the English Language says that [lutenənt] was recommended by Noah Webster in his American Dictionary of the English Language (1828). It thus seems to be a spelling pronunciation.
Wikitionary claims that leftenant is an archaic spelling of lieutenant.
My copy of the OED says:
The origin of the beta type of forms (which survives in the usual British proununciation, though the spelling represents the alpha type) is difficult to explain. The hypothesis of a mere misinterpretation of the graphic form (u read as v), at first sight plausible, does not accord with the facts. in view of the rare OF. form luef for lieu (with which cf. esp. the fifth c. Sc. forms luf-, lufftenand above) it seems likely that the labial glide at the end of OF. lieu as the first element of a compound was sometimes apprehended by Englishmen as a v or f. Possibly some of the forms may be due to association with LEAVE sb. or LIEF a.
In 1793 Walker gives the actual pronunciations as (lev- liv-tenant), but expresses the hope that ' the regular sound, lewtenant' will in time become current. In England this pronunciation (lju:'tenent) is almost unknown. A newspaper quot. of 1893 in Func's standard Dictionary says that (lef'tenent) is in the U.S. 'almost confined to the retired list of the navy'.
It's apparent from the examples of usage given that a mixture was in use, e.g. 1375 "lutenand" and "That..luf-tenand Was of to the king of Yngland", 1387 "leeftenaunt".
So it's clear variants of both were in use in England in the 14th century.
As to why present day usage is as it is: People can be contrary. It's possible the US adopted "Loo" because and only because the Brits said "Lef" -- or vice-versa. But it seems the answer is not known by the best scholars Oxford can produce.
While it will always remain a mystery, I think that this goes back to the OF pronunciation of "lieu" to sound like "lyeuch". Then "lieutenant" would have the pronunciation of "lyeuchtenant". Later, when the drive by the English to rid the language of french words began, they modified the word to try to match their pronunciation and made it "leftenant". Even later, when the Americans cuddled up to the French during the revolutionary war, their pronunciation changed to follow the french term. I'll bet George Washington used the English term at the beginning.
Of course- this is opinion and I have been wrong before. It was in 1950 I believe...
The different pronunciation started sometime after England was first at war with France from 1337 to 1453. The politicians and the public wanted nothing that sounded French. They also changed German Shephard dog to Alsatian.
However, we have been at war with the Dutch and still use some Dutch expressions like Dutch courage, go Dutch, Dutch yaw and Double Dutch.
Lieu from french shows up in English phrases like "in lieu of" meaning "in place of", so lieu means place and lieutenant essentially means placeholder. There is a rarely used word, lief, which I see in old books in phrases like "I'd as lief jump off this wall as ..." and I think it is another spelling and pronunciation of lieu. I think the "lef-tenant" pronunciation comes from that lief which was no doubt interchangeable with lieu at some point in the past.
BTW Canadians technically say lef-tenant though many ordinary folks say it the US way because that's what we hear on TV.
After the post, I will try to research the etymology of 'Lieutenant.' From what I was told (Fact or fiction as it may be), in the early Naval days (circa 1600), the sword of the Commanding Officer designated who was in charge and who was their leader. The Lieutenant always had someone guarding the sword when the Commanding Officer was wearing it. The sword is carried on the left hip. That person stood to the 'left' of the Lieutenant. Therefore, he was called the Left Tenant because he was second in command to the Lieutenant. If the Lieutenant was killed, the Left Tenant took the sword and became the Lieutenant.