Then you won't have a town that you own. You can still find a plot to live in someone elses town though.
Benefits: you get the mayor rank and all its benefits for free. Benefits include more locks for chests and such, being one step closer to the final rank (arguably the goal of main), your name is higher on the list when you press and hold tab, and (arguably) have a much better color next to your name (not to mention people tend to respect people who have a higher rank). Cons: you won’t own the town. Now, towns are unlimited, and if you upgrade to president you can get another “free” opportunity when you rank up again. Also, just because you don’t own a town doesn’t mean you can’t be a listed owner in another town (or even live in another town). It’s up to you tho
I personally would advise against it - The only real benefit here is a darker name and more LWC protections (which frankly I don't see you needing that many considering how early in the game someone who joined today would be). By forfeiting your Mayor town on the other hand, you lose your chance at getting a new town that's 100% owned by you, until you get President (which costs 180k on its own), which is arguably one of the important landmarks/goals in player progression in the game. Really up to you, but a fair cost-benefit analysis would lean against.
If I were you, and you wanted a fancy high-ranking tag, I'd save up for Mayor and then have someone buy you President 85,000 for Mayor for 1 town, and then $180,000 for President for another town. Town 3 would only cost you $220,000, which isn't much more than President would've cost, and you get a snazzy tag, as well as then having the ability to buy other Residents the Mayor rank in exchange fro their town if you ever want to expand. This is really only a good idea if you really want the President tag but... just earn it yourself otherwise
I gave both my Mayor town and President town away to rank up quicker, and whilst I don't regret it for mayor, I do for President. I'd say maybe only rank up once by giving your town away.