You could accomplish this several ways. First, since this is a web page, you could hide the text boxes on the page Load event if the dates are 1/1/1800.MyText.Visible = MyBO.MyDate.Year <> 1800
If you have any 3rd party web controls such as Infragistics that have a web based date time control, you can also set the "NULL" value to be 1/1/1800 which would show an empty field.
Last, you could create your own text box which and implement your own data binding and NULL value support. The NULL value in this case would be 1/1/1800 which you could then return an empty string when displaying the control.