The 24-year-old made 37 appearances, scoring five goals, to help the Tigers win promotion from League One.
"He will give us different options which will help us to evolve," said Hatters boss Nathan Jones.
"He turned down other Championship clubs to come here, so we are convinced of his commitment to become a top player, which we feel he will achieve."
Burke began his career at West Ham, making 15 appearances for the Hammers before joining Hull for an undisclosed fee in the summer of 2018.
He played over 100 games for the Tigers in total over three seasons.
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