No, they won't.

This paper examines the migration response to a millionaire tax in New Jersey, which raised the tax rate on top earners by 2.6 percentage points, becoming one of the highest rates in the country. Drawing on complete NJ state tax micro-data, we estimate the migration response of millionaires using a difference-in-difference strategy. The results indicate little responsiveness, with semi-elasticities mostly below 0.1. Tax-induced migration is higher among people of retirement age, people living off investments rather than wages, and potentially those who work (and pay tax) entirely in-state. The tax is estimated to raise $1 billion per year and modestly reduce income inequality.

(Thanks to Jake Blumgart for the study.)