Clashes broke out when efforts by riot police to stop the protest provoked demonstrators to throw flares and start fires.
Some protesters were frustrated that the international support for the damaged cathedral was eclipsing their demands.
“Millions for Notre-Dame, what about for us, the poor?” read a sign worn by a demonstrator.
“Everything for Notre-Dame, nothing for the miserables,” read another, referencing Victor Hugo‘s Les Miserables.
France’s interior minister said a total of 9,600 people were demonstrating across France, including 6,700 in Paris, in a largely peaceful day of protesting.
Many protesters were saddened by the fire at the cathedral, but they were also angry that some £770m had been raised so easily for the monument’s reconstruction while their demands remained largely unmet.
“I think what happened at Notre Dame is a great tragedy but humans should be more important than stones. And if humans had a little bit more money, they too could help finance the reconstruction work at Notre Dame. I find this disgusting,” said protester Jose Fraile.
Saturday’s protest marked the 23rd straight weekend of yellow vest action against economic inequality and Emmanuel Macron’s government, which demonstrators say favours the wealthy over ordinary workers.
Several thousand people marched from the Finance Ministry in eastern Paris to demand lower taxes on workers and retirees, and higher taxes on the rich.
Mr Macron was scheduled to lay out his responses to the yellow vest movement on Monday night, but his speech was cancelled after the fire broke out.
He is expected to rescheduled the speech for next Thursday.
The heavy police presence at the protests meant subway stations and roads around Paris were closed on Saturday, frustrating travel plans for tourists visiting the French capital.
Additional reporting by agencies