A big storm passed through our area and a bunch of trees fell. This one tree is kind of in a weird situation. The tree is technically on my neighbors property and it fell towards our house. However, the tree didn't completely fall and is being held up by another tree. If the other tree fails to hold it up, the main tree will hit our house. Who is responsible for removing the tree and would the removal be covered under insurance?
I've gone through this enough times in Pennsylvania to paraphrase the law in this state: If a tree falls it's nobody's fault, unless the tree's owner was given credible advance notice that the tree posed an exceptional risk of failure, in which case the owner is liable for damage it causes if it fails.
(For purposes of liability, a tree is "owned" by the owner of the ground where the trunk of the tree enters earth.)
In the scenario you describe: A tree owned by your neighbor has not yet caused significant damage, but has been so structurally compromised that it poses an exceptional risk to your property. The standard course of action in such a circumstance is to promptly and formally notify the owner of that fact, and for the owner to remedy the risk. (In this case, it sounds like the only practical remedy is removal of the tree.)
As a practical matter, the owner might be able to get an insurer to cover the cost of removal before it does more damage. But that's their problem.
As a matter of expedience, you might also notify your insurer, since if the tree does end up causing significant damage to your property, you could subrogate your claim through your insurer.
As a further matter of expedience: The township may have codes requiring landowners to address hazardous trees. Thus, if the owner does not promptly remedy the peril you could also notify the township.
You haven't said where this is. In jurisdictions I'm familiar with, property owners are responsible for trees on their property, so your neighbor is responsible for the tree and any damage it may cause. Whether your neighbor's insurance will cover the costs of removing the tree is between your neighbor and whatever insurers he may have.
If the tree causes damage to your property, and you're unable to recover anything from your neighbor, your insurance may cover some of your losses.