Law had joined the cast after Michael Fassbender left the Natalie Portman-starring, $15 million western, due to scheduling delays.
Law left the production because he had wanted to work with Ramsay and she was no longer directing the project.
"Warrior" filmmaker Gavin O'Connor has since replaced Ramsay as director, but Law's role hasn't yet been re-cast.
The Hollywood Reporter said there are conflicting accounts of why the production is having so many problems.
One scenario is that producers hadn't presented Ramsay with an approved schedule, script or budget just days before shooting was to begin this week. Another theory is Ramsay was revising the screenplay herself and failed to deliver a shooting script on schedule, the entertainment industry trade newspaper said.