To answer your first question, yes you can remove the tree roots that are under the liner. However, the more important question is whether or not the removal of the roots will cause short- and/or long-term damage to the affected tree. Removing a few roots, if done properly, should have limited impact on the the affected tree. However, I'm in no position to provide you blanket approval on such a tree root removal program without physically assessing the situation.
Therefore, my suggestion is that you need to contact a certified arborist for an onsite inspection and assessment. The Illinois Arborist Association maintains a list of certified arborists in the state of IL.