What is the ruling on someone who forgot to perform Sa'i during 'Umrah and then exited the state of ihram, and is expiation due upon him if he completes another 'Umrah after remembering?

Sa'y (the ritual circumambulation between Safa and Marwa) is one of the pillars of Umrah and is not waived due to ignorance or forgetfulness. Your Ihram for the last Umrah is not valid if you were still in the state of Ihram from the first one. However, if you performed Tawaf and Sa'y in the last Umrah, that suffices for the incomplete Umrah, and the rituals of Umrah you performed in the last Ihram are considered a completion of the first Umrah.