I have an editable Gridview with columns as below:
The only case that this happens is when you use UpdatePanel.
Now when you use UpdatePanel, on each UpdatePanel the Dom is change and the javascript need re-initializations. Now in your case you add new lines, and you make ajax update, so you need to re-initialize the date picker.
This can be done from the functions that come with UpdatePanel as:
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Asp.Net UpdatePanel in Gridview Jquery DatePicker
Datepicker for dynamically created controls