The incident, on Church Street, happened at about 16:30 local time on Tuesday.
A man arrested has now been charged by Gardaí (Irish police).
Gardaí said the fire was put out by Dublin Fire Brigade and the area made safe.
No injuries have been reported at this time.
The scene has been sealed off and a team from the Garda Technical Bureau were carrying out a forensic examination.
The man was arrested for an alleged offence under the Criminal Damage Act 1991.
'Disastrous for Dublin'
Archdeacon of Dublin and Vicar of St Michan's Church David Pierpoint said: "It's disastrous for Dublin, its disastrous for history and it's disastrous for the parish.
"The parish depends on income from our tourists and if we've no crypts to show people the various things that are down under the ground, then we have very little money coming in."
He said he would be contacting Ireland's National Museum to see if they could salvage something of the remains "but we're pretty certain that they've been destroyed" and are "beyond repair".
Church of Ireland Archbishop Michael Jackson said "the irreversible loss" was something that would "sadden generations, not only of people of Dublin and school groups, but international visitors who make a point of coming to St Michan's when they come to Dublin because they experience something unique
"People who were part of the Crusades were buried here and now it seems that option won't be open to them."
Crypt vandalised in 2019
In February 2019, vandals broke in and decapitated an 800-year-old skeleton known as the Crusader which was interred beneath the church.
The skull and another stolen from the crypt were later recovered.
The crypt was badly damaged in the 2019 break-in and several of the mummies - including the 400-year-old remains of a nun - were desecrated in the incident, according to the Church of Ireland.
What is the history of St Michan's?
The first church on the site is believed to have been established in 1095, but the current church dates back to the 1680s with further renovations taking place between 1723 and 1725, in 1767 and in 1825.
It was restored in 1998.
The vaults contain many mummified remains - as well as the almost 2m (6ft5) skeleton of the Crusader, they include the remains of brothers Henry and John Seares, executed by the English for leading the Irish Rebellion of 1798.
The remains of many of Dublin's most influential 17th, 18th and 19th Century families are also entombed in St Michan's.
The limestone walls of the crypts and atmosphere within them is believed to have helped preserve the remains.
The church's organ is housed within the case of an organ case constructed by John Baptiste Cuvillie around 1725.
Thousands of people visit the church each year.