Cutting tree roots
I am building a fence on the edge of my property. A 2 1/2' diameter (approximate) oak tree sits about two feet on the other side of the fence line, and I've run into a large root (about 5") within the first foot of one of the post holes. Will cutting this root be likely to damage the tree?
The roots of trees are important for providing the canopy with water, storing important minerals and sugars, and stability. Tree roots over 4 inches in diameter are likely to be structural roots; cutting these roots may impact the structural stability of the tree, creating the potential for the to fall over. The pictures would indicate this root is important to the stability of this tree. Since the tree is leaning away from the root, the root is acting like a weight balance. If you cut it, then there would be less support from the roots to balance the weight of the tree opposite of the way it is leaning. This could cause it to fall over in a strong wind.
Maybe the most important thing to consider here is if the tree falls could it cause structural damage and potentially hurt someone? It is also important to think about if it falls on your neighbors property would they hold you responsible. Long story short, I would do what you can to move the fence post away from the tree's roots system. As they grow they will continue to move the fence post around anyway.
Additionally, i noticed from the photo that there is English Ivy growing on the ground and up the tree. While you are working in the area, I would encourage you to remove this highly invasive species. It can also cause harm to the tree and other neighboring plants.