The saying 'levare le tende' meaning 'to leave' is quite common in informal Italian language. I guess its origin may come from the military jargon but could not find any evidence on this issue. I'd like to know if my assumption is correct or what other origin this expression has.
Italian word tenda is from late Latin tenda, meaning tent. In earlier Latin it was tentorium.
Typically, in ancient times, you would have nomadic tents or military tents. So, presumably, that is the origin of the idiomatic expression levare le tende (~ disassemble and remove the tents).
In La Bibbia di Gerusalemme the word appears several times, mainly in a nomadic and religious context. Itinerant people mount and unmount their tents all the time. In the Vulgata, the word tentorium has countless occurrences. The tenda del convegno (tabernaculum) was sacred. However I have not been able to find an Early Italian example where the expression was used in a nomadic context.
On the contrary I have found one occurrence in La Leggenda di Tristano (i.e. Tristano Riccardiano which was probably written in the early XIV century in Florentine dialect) where the context was military (levare le tende). An alternative expression which was used in the same context is levare lo campo.
(Anyway, my personal bias is that the origin was probably nomadic rather than military)