The Oxford English Dictionary cites the difference as coming from an Old French variant on the pronunciation of the word "lieu" itself. It is (rarely) attested as "leuf".

It's unclear how far back the split in pronunciation of "lieutenant" goes, but both "lieu" and "leuf" or "lef" spellings are attested in Middle English, so it's been around for quite a while, and likely has always been regional even within English speaking Europe.

Upon colonization of the Americas, one pronunciation predominated there, and the other in Europe, though it's worth mentioning that both are used in Canada, the "lef" pronunciation being considered more proper and used in governmental contexts (eg. Lieutenant Governor).