Can't quite tell what's going on here but, assuming your "rangelist" is
defined, why not just something like:

sub printem()
for each c in range("rangelist")
Range("TestName").Value = c
Sheet1.PrintOut
next c
end sub

There is an inherant paradox in using named ranges in code. A global named
range is visible from any sheet. When you write code however if you have a
line like

Range("NameList")
it is the same as
ActiveSheet.Range("NameList")

if NameList does not exist on the active sheet then the code will fail. So
as a guess your sheet 1 is active when you run this code. Since NameList is
on sheet 3 the code dies.

Take a look at this code to see if it makes sense to you...

Sub PrintMany()
   Dim rng As Range
   Dim rngToSearch As Range
   
   With Sheet3
   Set rngToSearch = .Range(.Range("NameList"), _
    .Range("NameList").End(xlDown))
   End With
   
   For Each rng In rngToSearch
       Sheet1.Range("TestName").Value = rng.Value
       Sheet1.PrintPreview
   Next rng
End Sub

or with your code...

Sub print_many()
 Dim iRow As Long
 iRow = 0
 With Sheet3.Range("NameList")
   Do
     Sheet1.Range("TestName").Value = Sheet3.Range("a" & iRow + 1).Value
     Sheet1.PrintOut
     iRow = iRow + 1
   Loop Until .Offset(iRow, 0).Value = ""
 End With

End Sub