I'm afraid not, no, its not even okay to graze them with the mower. The roots are a critical part of the tree, and if they are damaged, you may lose the tree. Welcome to GoY, George, sorry I've had to give you such a firm negative!

What kind of problem are they posing to you? Perhaps there is an answer where there will be no problem leaving them there.

Hi George & welcome.
How about if you cut out a circle around the tree base, removing the grass, then there would be no danger of you running the mower over the roots - it would look attractive.

Greenfinger, I suspect, by the look of the roots, they're some distance away from the birch, and that the birch has been in place for some time which means cutting a circle round it isn't really an option to solve the problem. But perhaps George will confirm or deny...

Hello George, are you there?

Yes,Loosestrife, I think you're right, you can see how the grass has died where the roots are.

Be careful if you think the roots are spreading towards your house. I had cracks in my garage wall due to a silver birch on the verge about 25 ft away. Up went my insurance premium to almost double and even though the tree has been removed I can't find a company that will set it any lower!