There is no reason to want to wash the lid in the dishwasher. At most, it builds up a little limescale, which can be removed with a little vinegar or limescale remover. The environment in the dishwasher is far too aggressive for plastic.
According to the operating instructions, the lid, the tea strainer and the jug may only be rinsed with water. Likewise, no abrasive cleaning agents or objects should be used. Therefore, a clear no to the dishwasher.