[Eomer of the Rohirrim]

I disagree, Morsul. Had the Ring been 'put to the sword', so to speak, Sauron would certainly have known, as he would have been reduced to some sort of ghost creature. The Ring was a great part of Sauron, but no great part of him was destroyed, so he can't have had any reason to think the Ring destroyed. He must have assumed it was lost, probably in the Great Sea. His fervent action came after he learned the Ring had been found, and wasn't lost anymore.

But to answer the original question, no I don't think Sauron suffered. The other Ringbearers suffered because the power of Sauron had corrupted them; Sauron himself could hardly be corrupted by his own power. Sauron's suffering came from the thought that a mighty Lord of Middle-earth could use his own power against him.