If there is an FFP IDIQ in place and making task orders from it is a Determination of Fair & Reasonable Price needed?
This is a good, interesting question. The short answer is no, you do not have to follow the procedures at FAR 13.106-3 However...
If you have an Indefinite-Delivery Contract you must follow the procedures at FAR 16.505 Ordering. For your scenario, the requirement at FAR 16.505(b)(3) Pricing Orders.
If the base contract (the IDIQ you reference) already established the prices for the supplies or services over the period of the contract; the fair and reasonable price determination documentation would have been accomplished at the time of award and does not need to be re-accomplished if issuing orders with the prices in the base contract.
Per FAR 16.505(b)(3), if the basic contract did NOT established the prices for the supplies or services, the contracting officer must establish prices for each order using FAR subpart 15.4. Yes, you would have to determine the price(s) are fair and reasonable prior to placing each order. See FAR 15.402(a).
FAR 13.106-3 would be applicable if you were using simplified acquisition procedures, which you would not be with an IDIQ.