There would be no damage, if they were in fact fired in a .30-06 chamber. I've seen lots of .270 Winchester brass that has been fired through .30-06 chambers and none of those rifles suffered any damage.
As for flame cutting, there wouldn't be any more flame cutting than a normal case firing in a normal chamber of the proper caliber. The case mouth is the case mouth, no matter what the caliber. This is an old wives tale, which if true, would mean you couldn't fire .22 Shorts, Longs and Long Rifles in the same chamber, or .38 Special in .357 Magnum chambers, et. al.
Hope this helps.
Fred