You may look at and play cards exiled face down with Dream-Thief's Bandana's first ability (and spend mana of any type to do so) even if Dream-Thief's Bandana and/or the equipped creature leaves the battlefield. If another player gains control of Dream-Thief's Bandana and/or the equipped creature, that player can't look at or play cards already exiled with Dream-Thief's Bandana, and you still can.