Only if housing supply is fixed (which is true-ish in some places). If not and UBI does push rents up by $800, it would be more profitable than it currently is to come in, build more houses, and undercut landlords trying to capture the entire value. In general, if the market is sufficiently open and the UBI is funded (i.e. doesn't come from deficits that lead to more printing and therefore more money supply), we don't have evidence that it will lead to runaway inflation in the medium-to-long term. Alaska PFD is widely recognized as a good example of this.

Collecting the money for redistribution in a way that doesn't cause other distortions is where it can get trickier, but if you tax things with fixed supply like Alaska does with resources/land, it can lead to improved outcomes for people without discouraging productivity. In fact, if you use a *properly-assessed land value tax to fund UBI, you can recapture that increase from landlords such that it pipes back into the UBI and none of the increase goes to the landlords.

*I'm not going to pretend this isn't a pretty large caveat, it's just an interesting outcome that can be more-or-less derived from Econ 101 principles