I've heard that we should not close our eyes while offering Salah or in sujood. Is it true?
As far as the state of eyes in sujood is concerned:
Ibn al-Qayyim said in Zaad al-Ma’aad that if a man can focus more fully on his prayer by opening his eyes, then it is better to do so. If he can focus more fully by closing his eyes because there are things that may distract him from his prayer, such as adornments and decorations, then it is not makrooh (abominable) at all and the view that in this case it is mustahabb (recommended) for him to close his eyes is closer to the aims and principles of shariah than saying that it is makrooh. (Zaad al-Ma’aad, 1/283).