• It makes it hard to move. Say you change jobs and now work in a different part of the city, or your kids move out, or you'd like to move in with a partner. Normally one of the benefits of renting is that you can move, but with rent control that means switching to a place that's much more expensive.
• Landlords benefit from having their long-time tenants leave. Without rent control landlords love long term tenants because they're reliable and they mean less work finding people for the apartment. Rent control reverses this, and the landlord loses the incentive to upgrade the apartment and otherwise keep the tenant happy. Yes, the landlord is being a jerk when they do it, but sometimes the best fix for widespread jerk-ness is changing the incentives.
• It keeps out outsiders. People who want to live in your city enough that they're willing to give up their existing local connections and start over in your city are really valuable, and rent control means they pay a lot more than people who've lived in the city longer.
• It depresses new construction. Yes, new buildings aren't subject to rent control, but in a place where everything is already built out you can't put up a new larger building without taking down an old one. Rent control means you either have legal restrictions saying you can't replace buildings, or you have protests against people doing this and displacing existing tenants. But in a growing city, without new construction housing is going to get more and more expensive.
• Rent control puts off dealing with the problem of housing costs. If everyone in was a renter and had to pay market rates there would be the political backing for changes that would make housing more affordable. Rent control means that the long-time community members who would be best at this political change don't really feel much urgency because they have relatively cheap places with rents set a decade or two ago. [1]

(These are pretty much the standard arguments against rent control; this isn't original to me. This is an expansion of a comment I posted earlier today in a discussion that had veered from FCC regulation into rent control.)

[1] Homeownership is also a problem here. Higher rents are another way of saying the land and housing are worth more, which means rents and property values move together. Homeowners wouldn't generally say they want people to have to pay higher rent, but they do want their homes to become more valuable, and this is another name for the same thing.