Exposed Tree Roots
There are no simple solutions. Sorry to say that no one can say for sure whether or not root removal will damage or kill the tree, especially from this vantage point. I would proceed with removal only if I was prepared to lose the tree. Chances are that the tree would survive but you're looking at a messy as well as laborious project, and then to learn later that new roots are growing back anyway. Adding soil to cover is another possible option if circumstances allow. You can read more about root issues at this link:
Otherwise an on site visit from a Certified Arborist would go a long way toward helping you make a decision. You can find one or more Certified Arborists in your zip code at www.treesaregood.com Click on Find an Arborist. Arborist evaluations are not expensive and he/she could also do the work for you if you desired, for another fee of course.