There are 23 stations along the northeast coast, and most of it is walkable, although some stretches are still under development. 

This page includes several walks around Durham.  CrossCountry runs trains along the East Coast mainline from Scotland to Southwest England.

This site includes information about walks in Northumberland. Northern Rail is the rail operator in along several lines in the North.

This tourism site includes a lot of walking information (not rail related).  Towards the bottom of this page, there are two walks along the Tyne Valley railway line in Northumberland.