I'm confused as to how auctioning rentals would distort the market.

All a landlord needs is one tenant who is willing to pay $N a week. An auction is just an efficient way of finding the appropriate value of N.

There are other issues which make auctioning rental properties a bad idea though. Auctions are good in situations where the appropriate price is unclear; in most rental markets it's fairly clear to everybody roughly what a given apartment is worth, though, plus or minus five or ten percent. The second issue is that as a landlord I'd much rather have a good tenant who I think is likely to be able to pay the rent than a tenant who looks bad on paper but is willing to pay an extra ten bucks a week; so I don't automatically want to choose the highest bidder as my tenant.