This Question has covered the recent news about a push for statehood for DC to redress the issues with lack of representation for residents.
What prevents people living in DC registering to vote elsewhere like the President recently did in Florida?
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Registering to vote requires residency in the state (or federal district) in which one is registering. States may vary in whether they require some form of proof of residency at the time of registration, or simply an affirmation. In either case, lying about one's residency status on registration forms would be fraud.
While some people have flexibility in where they register (university students, dual-residence retirees, a.k.a. snowbirds), most people do not.
Note that even President Trump's attempt to change his registration to Florida was initially rejected because he recorded his legal residence on the registration form as Washington DC, rather than Palm Beach, FL.
Every state only allows you to vote if you are a resident of this state.
When it comes to incumbent Presidents, their state of residency is generally considered to be the state where they were a resident before being elected, and where they intend to return once their time in office has ended. This is why Obama was able to vote in Illinois, and Bush in Texas.
In order for President Trump's voter registration in Florida to stand, he would have to convince authorities that he intends to move permanently to Mar-a-Lago (the address on his voter registration) once his time in office has finished.