"These are baseless allegations leveled to foment trouble,'' Prabhakar Tripathi, a spokesman for the central reserve police, said.
"In fact, we brought the fire engines to extinguish the fire."
Police said the shrine was only partially damaged and all its holy relics were safe.
But as the rumour spread, thousands of people took to the streets, chanting anti-India slogans and demanding Kashmir's freedom from Indian rule.
The protesters attacked a nearby police station, pelting it with stones, said Tripathi.
Police fired tear gas and used batons to disperse the protesters and at least three soldiers were injured in the fighting, he said.
Shiv Murari Sahai, Kashmir's police chief, said authorities were investigating the fire, but that it seemed to have been accidental.
The predominantly Muslim Kashmir region, divided between India and Pakistan but claimed by both, recently experienced some of the largest anti-India demonstrations in nearly two decades.
The protests began three weeks ago in Indian-controlled Kashmir when the state government transferred 99 acres of land to the Shri Amarnath shrine board, a trust running a Hindu shrine.
Kashmiris denounced the land transfer as an attempt to build Hindu settlements in the area and alter the demographics in the Muslim-majority state.
The state government later revoked the order.