Does Heedless Resurrection require you to pay the recycle cost of the unit you are playing from trash?

Heedless Resurrection does not require you to pay any "recycle cost."

The spell has an additional cost—killing a friendly unit—and it specifies that you play a unit from your trash that costs no more Energy and Power than the unit you killed. You do not need to pay the Energy or Power cost of the unit you are bringing back from the trash; the spell explicitly says to "ignore its cost."

Furthermore, "Recycle" is a specific game action (Rule 416) that is not part of the cost or effect of Heedless Resurrection.