It's possible that your company might march you out the door if you tell them you are leaving. I think this would be a real asshole move and not in their interest but it could happen. If this puts you in a major financial bind it might not be worth the risk, but if you are ok with this I think it is probably better for you to let them know.
I don't think you have an obligation to let them know but I think it's probably in your interest.