State Governor Gavin Newsom, a Democrat, signed the law last week.
In two lawsuits, the plaintiffs said it was unconstitutional as it added an extra requirement to run for president.
California is the first state to succeed in passing such a law, though others under Democratic control have tried.
It would apply to ballots for primary elections but not the general election.
It argues that the law would “directly impede” Mr Trump’s chances of gaining the Republican nomination. California provides 14% of delegates needed to win.
A second lawsuit by the Trump campaign argues that states do not have the power to supplement qualifications for president set out by the US Constitution.
Currently there are only three requirements for presidential candidates – to be natural born citizens, over the age of 35 and resident in the US for at least 14 years.
Mr Trump’s counsel, Jay Sekulow, described the law as an “attempt to circumvent the US Constitution”.
He said voters had already spoken about whether Mr Trump should release his tax returns.
However, Mr Trump is the first major-party presidential nominee not to release any tax returns since Republican Gerald Ford in 1976.