This is an open question in Colorado, where there's an ongoing challenge to the state's supermajority requirement in Kerr v. Hickenlooper. The state requires a supermajority for tax or spending increases, and several lawmakers challenged the law as violating the Guarantee Clause, which guarantees each state a republican form of government.
The case has been going on for a long time. It's about seven years old now, and they're still arguing about who has standing to bring the lawsuit. The latest development is that no one had standing, but that decision is currently being appealed in the Tenth Circuit.
A bigger problem than standing, though, may be justiciability. In Huddleston v. Sawyer, the Supreme Court of Oregon, rejected a challenge to supermajority requirements for reducing citizen-approved sentences for crimes, and in Lefkovits v. State Board of Elections, the Northern District of Illinois did the same with respect to judicial retention votes. Both courts concluded that supermajority requirements were a political issue, not a legal one.
The justiciability concerns should be tempered by two considerations. First, there's an argument in Kerr that because taxes and spending are such core legislative functions, that it should be harder to uphold restrictions on the legislature's ability to vote on them. Second, there's U.S. Supreme Court precedent in New York v. U.S. suggesting -- but not deciding -- that the questions about justiciability are overblown:
The view that the Guarantee Clause implicates only nonjusticiable
political questions has its origin in Luther v. Borden, in which the
Court was asked to decide, in the wake of Dorr's Rebellion, which of
two rival governments was the legitimate government of Rhode Island.
The Court held that “it rests with Congress,” not the judiciary, “to
decide what government is the established one in a State.” ... Over
the following century, this limited holding metamorphosed into the
sweeping assertion that “[v]iolation of the great guaranty of a
republican form of government in States cannot be challenged in the
This view has not always been accepted. In a group of cases decided
before the holding of Luther was elevated into a general rule of
nonjusticiability, the Court addressed the merits of claims founded on
the Guarantee Clause without any suggestion that the claims were not
justiciable. More recently, the Court has suggested that perhaps not
all claims under the Guarantee Clause present nonjusticiable political
questions. ... We need not resolve this difficult question today.
My money says that the question is not authoritatively answered in my lifetime.