Great question. As with other economic history issues, we're incredibly lucky that the Mesopotamian civilizations recorded their business transactions on clay tablets-- we've got much less from more recent civilizations, like the Romans-- most of their contracts have vanished.

We've got a lot of agricultural land rental agreements from various of these cuneiform tablets, and while much rarer, we do have house rental agreements as well. For example, there's the tablet BM17501 in the British Museum

This tablet, dating to the 17th year of Ammi-ṣaduqa, is a house rental contract for a property belonging to a high-ranking court official, the major-domo Riš-Marduk. He is known from other sources to be the son of the treasurer (šandabakkum) Marduk-mušallim. The renter is the judge Ipqu Annunitum, a well-known resident of Sippar-Amnanum. Riš-Marduk is not personally involved in the lease but has his interests represented by Akiya (a short form of the name Ikun-pi-Sin), the son of Ibni-Sin and the head of a very well-known family in the same town of Sippar-Amnanum commonly referred to as the ‘Akšaya family’ . Clearly Riš-Marduk’s main residence was in the capital city of Babylon but he employed Ikun-pi-Sin as his local agent to look after his property in Sippar. The contract, which confirms that Ipqu-Annunitum will pay the rent and keep the property in good condition, was part of Ikun-pi-Sin’s archive.

This is around the year 1600 BCE (that is, 3600 years ago), Sippar was a city to the Northwest of present day Baghdad. Note that both the landlord and the tenant are high status, this doesn't seem to be a rental that comes with any feudal obligations, it seems to be a fairly straightforward "I have this house, I rent it to you, you don't wreck it" transaction. Note also the use of an agent by the landlord to undertake the transaction on his behalf.

Its most typical to see houses as part of an agricultural rental -- hence "tenant farmers". Its rarer and less frequently encountered to see documentation of an urban tenancy in pre-modern times without a feudal relationship, but the legal format was there and these transactions did occur. Agricultural land rental agreements sometimes come with an obligation to build [rare], rebuild if destroyed [more common] or maintain [much more common] a dwelling.

Why do them? Earn income while keeping the asset-- that's a very attractive transaction. In today's London, many people -- including for a time, me-- are tenants (of a sort) of the Duke of Westminster's (Grosvenor family) Estate. Typically in a London rental the Westminster estate won't be the owner of the house, but rather the owner of the land on which the house sits-- this was the more common format, and it persists into the modern era. The Westminster estate typically leased land for 99 years, but in at least one instance entered into a lease to a Council for 999 years -- you'll often see UK homes listed by years remaining in the ground lease. This approach is disfavored in American law and real estate practice-- our preference is to purchase "in fee simple", that is outright; you encounter these long leases in commercial real estate settings more often in the US.

But in a world where a few powerful people controlled much of the land and could dictate the terms of transactions, why not hold onto the property title, if you can? The Dukes of Westminster could, so they did. While the Grosvenors don't know what the world will look like in 999 years or that they will be around, being a landlord gives them the power to enforce covenants and protect other interests in ways they could not if they'd sold outright.

Sources:

A Discussion of Ground Rent

"The Scribe of the Flood Story and his Circle", in: Karen Radner and Eleanor Robson (eds.), The Oxford Handbook of Cuneiform Culture

Westminster City Council v Duke of Westminster

Settled Land. Tenant for Life. Building Lease for 999 Years under Settled Land Act, 1925, s. 41. Covenant by Lessee to Rebuild, No Definite Time being Specified for This. Covenant Held Valid. Settled Land Act, 1925, ss. 41, 42, 44, 117. Re Grosvenor Settled Estates; Duke of Westminster v. McKenna. [1933] 1 Ch. 97