Do I need to recycle a purple rune to equip Last Rites on play?

Yes, you must recycle a purple (Chaos) rune—or any other rune—to pay for the cost, but specifically, you must recycle a Chaos rune (or use Universal Power) to pay the [C] (Chaos Power) requirement within the Last Rites equip cost.

According to Last Rites' card text, its [Equip] ability cost is: [C] + Recycle 2 cards from your trash.

Here is the breakdown of why you must recycle a rune:

1. The [C] Cost: This requires 1 Power of the Chaos (purple) domain. As per Rule 162.2, Power costs must be paid with Power of the appropriate domain. You generate this by recycling a Chaos rune (or using Universal Power if available) as a [Reaction] during the "Pay the card's costs" step (see Rule 357.1.a and RiftJudge FAQ #4296).
2. The Recycle Cost: You also must perform the action "Recycle 2 cards from your trash" to satisfy the second part of the Equip cost. Per RiftJudge FAQ #8181, you cannot activate the Equip ability unless you have at least two cards in your trash to recycle.

Summary: You need to provide 1 Chaos Power to satisfy the [C] cost (typically by recycling a Chaos rune) AND you must recycle 2 cards from your trash to satisfy the "Recycle 2 cards" portion of the cost. You cannot equip Last Rites without paying both of these components in full.