The Constitution applies only to state action. Thus, if the landlord is a person or private entity, no right of action exists based on constitutional claims. The exceptions here, instances in which private action must comply with constitutional norms, are public function and entanglement. I doubt public function would apply since it appears only that there is an issue of managing property and not, e.g., running a town or hosting an election (activities "traditionally and exclusively" performed by the state). Entanglement would only apply if the lease execution by the private fee owner is somehow being facilitated or authorized by the government. If I recall correctly, this concept is at issue in Rendell-Baker v. Kohn, 457 U.S. 830 (1982) - private school teachers fired over speech sued the school alleging state action due to public subsidies (amounting to 90%+ budget) and performance of a government-related contract. Teachers lost.... difficult burden to prove state action.
If the government is the lessor, I still don't see significant issues. If the government is renting or leasing property to persons or private entities, it is creating a private property right for such lessees for the duration of the agreement. Free speech does not seem to be an issue since there is no right of action for 1st amendment rights within private property. It seems difficult to rationalize the choice of tenant as ex-ante regulation of speech. Since there's no protected class at issue, the process of choosing a tenant also does not appear to pose any constitutional problems.
On the other hand, if your scenario contemplates something like a public stage where persons or groups pay something of a "permit" fee for its use, then the discussion may turn to content and viewpoint based restrictions - and perhaps prior restraint issues associated with the permitting/licensing. But I'll stop here for now.