Can you bin a card to madness into exile and then cast it for its madness cost?

When you discard a card that has Madness, a replacement effect applies: instead of going to your graveyard, the card is exiled face-up. This is covered by CR 702.35b, which states that if a card with Madness would be discarded, it is exiled instead.

Once the card is in exile (having been discarded), its Madness triggered ability triggers and goes on the stack. When that trigger resolves, you may cast the card from exile for its Madness cost rather than its normal mana cost. If you choose not to cast it, it then moves to your graveyard (CR 702.35c).

It is important to note that the card never actually touches the graveyard during this process — it goes directly from your hand to exile. The term 'bin a card to madness' is colloquially accurate: you discard it, it exiles, and you cast it from exile.

Example: You control a land that says 'Discard a card: add one mana of any color.' You discard Fiery Temper (Madness {R}). Instead of going to the graveyard, it is exiled. Madness triggers; when the trigger resolves, you pay {R} and cast Fiery Temper from exile, dealing 3 damage to any target.

If you choose not to cast the card when the Madness trigger resolves, it simply moves to your graveyard at that point and cannot be cast for its Madness cost later (CR 702.35c).