The line, which travels through Darlington, York and Doncaster, will be closed between London and Peterborough until at least noon today as urgent repairs are carried out, with a reduced service planned even when it does reopen.
LNER urged passengers not to travel, with those holding tickets today told they will valid for use tomorrow and Friday.
An LNER spokesperson said: "Following the severe weather and high tempeatures on Tuesday 19 July, a fire near the railway south of Peterborough has caused significant damage to signalling equipment. Network Rail are working to repair this as quickly as possible, but the extent of the damage means that services will be disrupted for the duration of today, Wednesday 20 July.
"There are numerous reports of damage to tracks, signalling equipment and overhead lines across the UK rail networks, including the routes we would typically redirect our customers to. With that in mind, unfortunately it will not be possible to complete your journey with another operator via these routes today."
Dozens of trains are being cancelled or delayed across England on Wednesday morning because of problems caused by the extreme heat.
Temperatures hit 40C on Tuesday, the hottest on record in the UK, causing damage to overhead wires, tracks, and signalling systems.
National Rail has told customers to check before setting off on their journeys and to only travel if absolutely necessary.
It said on its website: “A huge amount of work has been completed overnight to fix problems with the track and overhead wires caused by the heat yesterday.
“Network Rail teams are continuing to work tirelessly to make the repairs so we can get services back up and running for passengers but there is still disruption to services throughout the day. So, for anyone travelling today, please take time to check before you travel.”
More than two dozen services departing from King’s Cross and Euston have been cancelled or delayed.
Provider Avanti West Coast said services from Euston were cancelled due to ongoing repair works following damage caused by fires in the Hatch End area on Tuesday.
All lines between Lancaster and Carlisle and lines in the Birmingham New Street area were blocked due to severe damage to the overhead electric wires.