Journal Batch Mode : Error

Is NX running with a loaded part when you run your script?

Looks like you don't have a displayed part.
So dp in line 12 will be "Nothing".
Then, when you try to get dp.Notes in line 13, you get a NullReferenceException

I think that you will need to pass in the part you want to edit as a parameter to the journal. The journal will then need to open this part and act on it. Call the code below in the NX command prompt with this command:
run_journal {path to ReplaceString.vb} -args {path to part}

I've added some code to open and save the part (if you do not save the part in batch mode, you won't see the changes); the added code does no error checking, you'll want to make it more robust for actual use.

Option Strict Off
Imports System
Imports NXOpen
Imports NXOpen.UF
Imports NXOpenUI
Imports NXOpen.Utilities
Imports NXOpen.Annotations

Module ChangeNoteWord
Dim s As Session = Session.GetSession()
dim lw as listingwindow = s.listingwindow
Dim dp as part
Dim wp as part

Sub Main(args() as string)
lw.open
OpenPart(args(0))
Dim nc As NoteCollection = dp.Notes
Dim notetext1 As String = "existing word"
notetext1 = "COOL"
Dim notetext2 As String = "new word"
notetext2 = "TEST"
Dim notestring() As String
Dim nolines As Integer = 0
Dim found1 As Boolean = False
Dim m1 As Session.UndoMarkId = s.SetUndoMark(Session.MarkVisibility.Visible, "M1")
For Each a_note As Note In nc
notestring = a_note.GetText()
nolines = notestring.Length
For i As Integer = 0 To nolines - 1
found1 = notestring(i).Contains(notetext1)
If found1 = True Then
notestring(i) = notestring(i).Replace(notetext1, notetext2)
End If
Next
a_note.SetText(notestring)
Next
s.UpdateManager.DoUpdate(m1)

Dim partSaveStatus1 As PartSaveStatus
partSaveStatus1 = dp.Save(BasePart.SaveComponents.True, BasePart.CloseAfterSave.False)
partSaveStatus1.Dispose()

End Sub

sub OpenPart(byval thePart as string)
Dim basePart1 As BasePart
Dim partLoadStatus1 As PartLoadStatus
basePart1 = s.Parts.OpenBaseDisplay(thePart, partLoadStatus1)

dp = s.Parts.Display
wp = s.parts.work

partLoadStatus1.Dispose()
End Sub

Public Function GetUnloadOption(ByVal dummy As String) As Integer
GetUnloadOption = NXOpen.Session.LibraryUnloadOption.Immediately
End Function
End Module