Any time you intentionally injure, excavate, or remove tree roots, you run the risk or introducing unintended consequences; however, if the tree is healthy the probability of long-term damage is low. I would suggest you first speak with your neighbor to determine how intrusive the root is on their property. Secondly, I would contact your local city government to see if they have any ordinances regarding trees/vegetation. Most local ordinances I am familiar with allow neighbors to prune vegetation that extends onto their property, but only if they do not injure/kill your living tree, shrub, etc. as a result of such action.
Best of luck!
