As stated by others, the local laws may provide otherwise, but you should also check with your insurance company.
In most cases, as I've been advised by my insurance, you would be liable for any damage done by your tree to your neignbors property.
You are not required to remove any limbs that extend over your neighbor's property, but your neighbor may remove them if he wants.
If you pay to remove them, you will probably be liable if there is a mishap. Why not tell you neighbor to pay to remove them, and his contractor would then be responsible.