Is a Muslim required to remember whether he has prayed previous prayers, such as Dhuhr or Fajr, when performing the Maghrib or Isha prayers, or not?

It appears that what the questioner is experiencing is a form of waswasah (whispers/obsessive thoughts), and he should disregard it. It is obligatory for a Muslim to perform prayer at its prescribed time, and he is not required to ponder whether he has prayed it or not. However, if he is not afflicted by waswasah and remembers that he has not performed a prayer, then he must make it up.