I booked an award four days out for a flight last week, didn't get asked for the payment card (booked on my card, me flying).
Three years ago, I booked a revenue ticket using a supplementary card, and got asked for sight of the main card at checkin - so as Craig says, YMMV.
I'd suggest that if you're worried, go to an Iberia sales desk at an airport on your travels between now and flight time, and show them the actual card - they don't need to see it at checkin, they just need to have seen it before they will check you in.
95% you wouldn't have any issue anyway, but that would cover the 5%, if it's not too onerous.