Network Rail had to pay out almost £60,000 in compensation when 59 train services were delayed by people encroaching on the track to get as close as possible to the iconic engine as it travelled from London to York in February.
Trains on the East Coast Main Line were temporarily stopped, causing a combined total of more than eight hours of delays.
Timings of the run will not be released to deter enthusiasts.
Network Rail managing director Phil Hufton said: “While the turnout to see Flying Scotsman so far has shown the passion and support for steam engines, and indeed the railway itself, the images of people stood on the railway taking photographs were deeply concerning and a breach of our safe operations,” he said.
“I cannot stress enough how dangerous it is to go onto the railway without any formal training and without permission, as well as being illegal.
“I am urging those who plan to enjoy seeing Flying Scotsman in the coming days to do so from a safe position and do not go onto the railway under any circumstances.”
The National Railway Museum in York bought the locomotive for £2.3 million in 2004 before work got under way on its restoration in 2006.
It is not publishing the timings of its tour in a bid to reduce the chance of people disrupting mainline services.
Jim Lowe, head of operations at the museum, said: “It is vital that spectators do not venture onto the railway, particularly when it is on the mainline as a full timetable of regular services will be running.
“In order to avoid overcrowding and incidents of trespass neither ourselves nor our partners will be publishing recommended viewing points or the timetable of when the train will be passing through specific locations.”
Flying Scotman was built in Hexthorpe, Doncaster, in 1923. It pulled the first train to break the 100mph barrier in 1934.