There is a new study just released that shows there is a link between losing a lot of trees in your neighborhood and declining health of the residents in that area. It is a fascinating read, and it underscores the critical importance of not bringing in firewood from far away (which can start an infestation that then decimates the trees) and also reporting any odd pests or damage quickly, so that you might lose one tree instead of all of them.
One thing I think gets a little lost in some of the other reporting of this article is that the emerald ash borer really has nearly nothing to do with the big picture. The scientists used neighborhoods that had been infested with EAB not because there is something particularly dangerous or bad about EAB, but because it is a pest that kills a lot of trees in many places. This same study could have easily been done with neighborhoods affected by Dutch Elm Disease in the 1930's and 1940's, for instance (had the technology and information been there for the research, of course). Likewise, it is important to remember that control methods for EAB, ALB, and other insects and diseases remove trees from neighborhoods that are going to lose all their trees no matter what. It isn't like ignoring the pests and dead and dying trees results in less damage. In fact, the most direct result of not controlling pests is the loss of more trees over the long run.