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.

You pay all costs and follow all normal timing rules for cards played from exile with Dream-Thief's Bandana's first ability. For example, if the exiled card is a land card, you may play it only during your main phase while the stack is empty.

Playing an exiled card causes it to leave exile. You can't play it multiple times.

If you leave the game, any remaining face-down exiled cards remain exiled face down indefinitely. No player may look at them.