Does exiling a card with flashback from the graveyard prevent it from being cast with flashback?

Flashback grants an alternative cost that lets you cast a card from your graveyard. The ability specifically requires the card to be in the graveyard at the time you cast it. This is defined in CR 702.34a: 'Flashback [cost] means: You may cast this card from your graveyard by paying [cost] rather than paying its mana cost.'

Once a card is exiled, it moves to the exile zone and is no longer in the graveyard (CR 400.1, 400.2). Because flashback can only be activated from the graveyard, a card that has been exiled is simply not in the correct zone to use the ability — the permission never becomes available.

Additionally, flashback itself includes a replacement effect: if a spell cast with flashback would go anywhere other than exile after resolution, it is exiled instead (CR 702.34b). But this is only relevant after the spell resolves, not before casting. If the card is already in exile before you try to cast it, flashback never applies.

Example: Your opponent controls Rest in Peace, which exiles cards that would go to any graveyard. When your Faithless Looting is exiled by Rest in Peace rather than going to your graveyard, you cannot later cast it with flashback because it is in the exile zone, not the graveyard.

The same logic applies to any effect — such as a Tormod's Crypt activation or a Relic of Progenitus trigger — that exiles the flashback card directly from the graveyard before you have a chance to cast it.