Reiterating: I am not a lawyer. :)
The standard used throughout the legal system is the mythical "reasonable person".
Dead trees are known to drop branches and fall down, so completely ignoring a request to prune and/or remove a dead tree that somebody has expressed concerns about is...
A tree that has a healthy branch that comes off due to a severe storm is probably not the liability of the tree owner. A tree that is a clear danger but which the owner does nothing to remedy is almost certainly the tree owner's liability if it causes damage to people or property. Everything...
Disclaimer: I am not a lawyer.
If the neighbours apply for a tree order, and the tree is dead, I would think they're likely to get one. You can either waste court resources by forcing them to get one, or just trim / remove the tree...