The ASP.NET code for update panels in .NET 2.0 only works with browsers that does not recognize the new HTML5 input type - it will only do partial postback of known old fashioned input types.
I think there was an update for .NET 4.0 that might fix this - so try upgrading your project and see if it then works.
Otherwise you have to use and old style input field and code the date selection yourself, or add javascript to copy the input from the new HTML5 fields to a hidden field before the ASP.NET update panel javascript logic fires.
(You might also find that the implementation of date input varies too much between browsers to be of any use yet, so consider adding your own date selection UI optimized for the kind of date input that you would like.)