What would happen if someone filed a Writ of Quo Warranto?

A writ of quo warranto is not a petition, but a notice of demand, issued by a demandant, to a respondant claiming some delegated power, and filed with a court of competent jurisdiction, to hold a hearing within 3 to 20 days, depending on the distance of the respondant to the court, to present proof of his authority to execute his claimed powers. If the court finds the proof insufficient, or if the court fails to hold the hearing, the respondant must cease to exercise the power. If the power is to hold an office, he must vacate the office.

The writ is unlike a petition or motion to show cause, because the burden of proof is on the respondant, not on the demandant.

By itself, the writ does not seek the support of the court to order the respondant to cease the exercise or vacate the office. That would be an accompanying writ of prohibito or a writ of mandamus. All such writs contemplate enforcement by the people as militia, although that could include the sheriff or constable as commander of militia. The right involved is that of the respondant to present his evidence.

These writs are called prerogative writs because they are supposed to be docketed ahead of all other cases except other prerogative writs. The demandant represents the sovereign, the people, and anyone may appear in that capacity, even without a personal stake in the decision.

A writ of habeas corpus may be regarded as a subset of quo warranto, for cases where the claimed power is to hold a prisoner, but with the addition of a requirement to produce the prisoner in court, not just appear to present evidence of authority.

The prerogative writ of quo warranto has been suppressed at the federal level in the United States, and deprecated at the state level, but remains a right under the Ninth Amendment, which was understood and presumed by the Founders, and which affords the only judicial remedy for violations of the Constitution by public officials and agents.

https://constitution.org/1-Activism/writ/quo_warranto.htm