The fire happened between junction 5 near Sevenoaks and junction 6 for Godstone at around 10pm on Sunday.
All lanes in a clockwise direction are closed.
National Highways said it expected the road to stay shut “throughout the day” and that delays were around two and a half hours with four miles of congestion.
A diversion involving A-roads is in place.
“Specialist recovery will be required and the road surface will likely need to be resurfaced,” it added.
Surrey Fire and Rescue said it sent seven appliances to the scene when the blaze broke out.
The incident has also shut the M26 westbound between junction 2A and the M25/A21.
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Separately, the northbound tunnel of the Dartford Crossing between Essex and Kent is closed because of a “large spillage” creating severe delays.
National Highways said overnight roadworks overran after a maintenance vehicle broke down and emergency resurfacing works were needed.
It said the tunnel is expected to remain closed throughout the day and delays were about 90 minutes.
