VBA to refresh Pivots

You only want to display the error message if there is an error. Additionally, you may wish to check if the error happened while the pivotTable object was assigned:

Sub RefreshAllPivots()
    On Error GoTo ErrHandler

    Dim pt As PivotTable
    For Each pt In ActiveSheet.PivotTables
        pt.RefreshTable
    Next pt
ErrHandler:
    If err Then
        If Not pt Is Nothing Then
            MsgBox "Error Refreshing " & pt.Name
        Else
            MsgBox "Unexpected error"
        End If
    Else
        MsgBox "All Pivots Refreshed"
    End If
End Sub

Note that I renamed your pivotTable variable to pt - it's not great practice to use reserved words as variable names.

The error "object variable with block variable not set" is thrown, because pivotTable.Name is not declared after the For Each pivotTable In ActiveSheet.PivotTables loop.

This variable is assigned to a value only within that loop. The Exit Sub before the error handler is a best practice in VBA:

Sub RefreshAllPivots()

    On Error GoTo Errhandler

    Dim pivotTable As pivotTable
    For Each pivotTable In ActiveSheet.PivotTables
        pivotTable.RefreshTable
        Debug.Print pivotTable.Name
    Next

    MsgBox "All Pivots Refreshed"
    Exit Sub

Errhandler:        
    MsgBox "Error Refreshing " & pivotTable.Name

End Sub