How many millimeters of a front natural tooth's surfaces is removed to then add bonding to close a narrow gap?

Some to none: Wow, tough question. I have found that i need .75 to 1.25 mm to provide natural color and shape to a tooth that i restore with a veneer. However when closing a gap there can be less tooth removed, because I am adding restorative material there. It looks like a lot of tooth being removed but it all turns out fine in the end. I would expect that you are having crowns done which is 1-1.5 mm deep.

Close the gap.: Typically, for the front teeth, it's not necessary to remove any tooth structure when closing a gap between the teeth unless there is an existing filling or decay.