Serbs declared the creation of their own state within Bosnia on 9 January 1992, fuelling an ethnic conflict in which about 100,000 people died.
Bosnia is still split along ethnic lines between the mainly-Serb entity and a Muslim-Croat federation.
The Constitutional Court, based in the Bosnian capital Sarajevo, banned the referendum.
Polls opened at 07:00 local time (05:00 GMT) and will close at 19:00.
About 1.2 million people are eligible to vote.
As well as being the anniversary of the declaration of a Bosnian Serb state, 9 January is an Orthodox Christian holiday.
"The Republic is going into a referendum. It`s a great day for our Republic and our people," Mr Dodik said on Friday.
"We have to show our dignity, that we are a democratic people and that we have the right to make our own decisions."
But Bosniak leader Bakir Izetbegovic has accused Mr Dodik of "playing with fire".
The BBC`s Guy Delauney in neighbouring Serbia says that the Bosnian Serb leader has been deliberately provocative by insisting on celebrating the national holiday on 9 January.
Analysts say by flouting the court, one of the federal institutions set up at the end of the war 21 years ago, Mr Dodik is threatening the Dayton peace agreement, which ended the war in 1995.
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