If the text is not found then the object FoundOne will be nothing so you need to test for that first.

    Dim DXFDict, fAddress As String, LookInR As Range
    Dim DXFDescription, DXFCode, Key, FoundOne As Range
  Set DXFDict = CreateObject("Scripting.Dictionary")
   DXFDict.Add "ROAD", "10"
   DXFDict.Add "TREE", "24"
   DXFDict.Add "FENCE", "30"
   DXFDict.Add "GATE", "31"
   'DXFDict.Add "", "0"
      Set LookInR = Sheets("Orienteering").Range("E:E").CurrentRegion
          For Each Key In DXFDict
              With LookInR  'Columns("E")
                 Set FoundOne = .Find(What:=Key, LookAt:=xlPart)
                 If Not FoundOne Is Nothing Then
                    fAddress = FoundOne.Address 'ERROR ON THIS LINE
                      Do
                        FoundOne.Offset(1).Value = DXFDict.Item(Key)
                       Set FoundOne = .FindNext(After:=FoundOne)
                     Loop While FoundOne.Address <> fAddress
                 End If
               End With
          Next