Chestnuts are on the non toxic list of plants so they should be okay, however eating many of them can cause stomach upset . Nuts in general are fatty which can cause a dog to get pancreatitis.
Here is the non toxic plant list; http://www.sniksnak.com/plants.html
Toxic plant listing http://www.sniksnak.com/plants-toxic.html
Horse chestnuts can cause a problem though. http://en.wikipedia.org/wiki/File:Horse-chestnut_800.jpg
Regular chestnuts http://en.wikipedia.org/wiki/Chestnut