Yes being tired definetely has a huge impact on lucid dreams, on two fronts. The first being that (My opinion, could vary) the brain is more tired and has less time for awareness in the dream. The real problem however is that if you wake for WTBT or try to record your dream journal you will likely do it wrong or forget or fall asleep. If you are not aware enough for WBTB to bed you usually fall asleep again (Which is probably the better option if you hadn't had much sleep) and when you often forget to record dreams the moment you wake up losing valuable information.