Who determines whether it is a clear danger? An annoyed neighbour might say its 'obvious'. The tree owner might say its fine. An arborist would know but would the tree owner be obligated to get one anytime a neighbour raises the question?
If it hasn't dropped any branches before, is there any reason to suspect a tree would start? Pardon my ignorance here, is a dead tree highly likely to drop branches? If so, then I would agree that the neighbour is justified in requesting it be cut down.
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 likely to be found to be unreasonable.
If neighbours complain that they have concerns that a tree of yours is going to injure them, then it's up to you how comfortable you are with risk as to what you do. If you do nothing - not even investigate - and the tree injures them, then you're highly likely to be found to be liable and find the judge very unsympathetic.
If you get an arborists' report and the arborist says there's a very low risk of injury, then you'd probably be found to have acted reasonably in not doing anything.
If you do nothing and no person or property is injured, then no problem - at least in negligence.
But putting aside the issue of negligence - where your property (tree) damages the property of others, or injures them - there's also a tort of private nuisance, which is where your property interferes with the amenity of others.
If the neighbours say that they're afraid of your tree, and their fear is deemed "reasonable" (i.e. others are likely to share that fear), then it might reasonably be found to be causing a private nuisance, and you may be obligated to remove it, whether it's dangerous or otherwise.
I recall an example where a crane engaged in construction work was overhanging a neighbouring house. The neighbour went to court because she said she was fearful that the crane might fall on her house, even though it complied with all safety regulations. She won, and got an injunction preventing the crane from protruding over her house, and I think she got compensation, too.
I can't see why the same principles wouldn't extend to an overhanging tree. (Noting that the interference, in private nuisance, has to be "substantial and significant", so a few leaves and twigs certainly wouldn't constitute a nuisance, a very heavy limb over a house very well might.)
