The main suggestion is to create the form as a template instead of a
document, and use File > New to create a new document based on that template
for each use. That way the fields will always start out empty, and you don't
need a button.

The alternative is fairly nasty. Create a macro like this (assuming all form
fields are text, not dropdowns or checkboxes):

Sub ClearFields()
   Dim fld As FormField
   For Each fld In ActiveDocument.FormFields
       fld.Result = ""
   Next
End Sub

Insert a macrobutton field,

  {MacroButton ClearFields Double-click here to clear fields}

(see http://www.word.mvps.org/FAQs/TblsFldsFms/UsingMacroButton.htm).

To prevent the macrobutton field from printing, surround it with a
bookmark -- say you name it ClearButton. Then write two macros to intercept
the File > Print menu command and the Print button on the toolbar
(http://www.word.mvps.org/FAQs/MacrosVBA/InterceptSavePrint.htm):

Sub FilePrint()
   With ActiveDocument
       If .ProtectionType <> wdNoProtection Then
           .Unprotect
       End If

       .Bookmarks("ClearButton").Range.Font.Color = wdColorWhite
       Dialogs(wdDialogFilePrint).Show
       .Bookmarks("ClearButton").Range.Font.Color = wdColorAutomatic
       .Protect Type:=wdAllowOnlyFormFields, NoReset:=True
   End With
End Sub

Sub FilePrintDefault()
   With ActiveDocument
       If .ProtectionType <> wdNoProtection Then
           .Unprotect
       End If

       .Bookmarks("ClearButton").Range.Font.Color = wdColorWhite
       .PrintOut Background:=False
       .Bookmarks("ClearButton").Range.Font.Color = wdColorAutomatic
       .Protect Type:=wdAllowOnlyFormFields, NoReset:=True
   End With
End Sub