So lots of people believing the rich are extra mobile and that they will leave if their [local] taxes are raised doesn't apparently make it so. At some level the argument that high taxes drive rich people away is, on its face, ridiculous. Look at really highly taxed places in this country--NYC, San Francisco, NJ--and guess what, more rich people live in those places than live in the low taxed places in this country (both in terms of raw numbers, and, more relevantly, proportionally). So clearly low taxes aren't very useful for retaining rich people.
I would argue that the very things that high taxes can provide: better education, more/cleaner parks, better security...are reasons that all people would want to live in those places, but since many people want to live there, those places become expensive to live in and, over time, more wealthy people end up living in them than poorer people.
That said, it isn't necessarily the case that places considering a millionaire's tax shouldn't worry that their millionaires will leave. It's more complex than just what we currently see. If wealthy people living in low tax states are living there in part because they are more inclined to go to low tax areas, then they will move if their taxes go up. In the case of Florida, while they could probably get away with it, it may also be that Arizona, New Mexico and Texas would start to see a larger share of the rich who do move.
If you pay the same taxes in Arkansas as you do in Manhattan, well, New York has lots more cultural opportunities, more diverse food/entertainment, better access to airports that will shuttle one to the far corners of the world, nearby ocean, mountains, rivers...also, it has more rich people already, and the wealthy do love rubbing elbows with each other. Arkansas does have Petit Jean State Park, which is gorgeous, but probably isn't enough to compete.
So while there's a pretty good argument that, in general, high taxes on wealthy people are not going to lead to an exodus of those same people, it is probably quite dependent on what taxes are already, where the increase would put them, and the particulars of a state/region and the rich that choose to live there.