'Lever' doesn't make any sense. Are you sure that he didn't say "The lift takes forever"? As your first language is Chinese, it seems likely that what you heard as a letter L was actually an R. If you were both waiting for the lift to arrive, that would be a logical remark to make. It means that it usually takes a very long time to arrive. When two strangers find themselves together with nothing to say, it is normal to fill the silence with a remark about the situation that you have in common, whether it's a delayed bus or cold/hot/rainy weather. In this case, a comment about how slow the lift is would be very normal.

As Su.Ki says, The lift (or elevator) takes forever is exaggeration of "The lift (or elevator) is slow." Lift versus elevator is just a regional difference. Most places will understand both and some will make conversation of the difference.

When he said a lift, I assume he meant an elevator. Even though an elevator isn’t technically operated by a level, and instead by a pulley, the phrase “a lift takes a lever” has better alliteration with two words that start with the letter ‘L’. So the man most likely was intending to make a point about how even though we don’t see it or appreciate it, the elevator has something doing the work for it. If he was talking about a car ride (which is what I first thought, although now that I’m reading the question again it seems like it’s just an elevator), that man was probably using the word ‘lift’ for both the elevator and the ride he was about to take—insinuating that the ride is like an elevator, as while it gets you to where you need to go, we sometimes don’t appreciate the effort made to get there. This would have been him recognizing the effort being made by who was driving him. If it’s just an elevator he’s waiting for, it was probably just him recognizing the hard work that goes behind the mechanics of an elevator.

when he said? moment after you enter into lift or after sometime?