You can re-enter as a tourist with no NOC with a US passport. It just depends on whether your new employer wants to do the 'right' thing and have you re-enter Qatar on business visas for your business trips (in which case an NOC would probably be required) or content for you to come in to do work on a tourist visa. I've known people who have lived and worked in Qatar for years on tourist visas, so I can't see it being a problem.