To 'run your eye over something' means to take in the whole of something or to look quickly at something. When you 'throw your eye over' something you look at it without giving much attention to detail usually in order to approve something or to give your opinion. Similar in meaning to 'cast your eye over'.

1. Chris ran his eye over John and knew immediately that he wasn't feeling well, so he only stayed for a short while. 
2. Would you mind throwing your eye over my homework? I want to make sure there are no mistakes.
3. I asked Helen to throw her eye over my painting because I really valued her opinion, and knew that she would have no hesitation in speaking her mind. 
4. She ran her eye over the room and immediately knew something was missing.

1. Kate, would you mind throwing an eye over this report before you leave? I just want to make sure that I'm on the right track with it.
2. I like to delegate responsibility and leave my staff to a task, but occasionally I will come into the office to run my eye over their work and see if they're having any problems.