There are a couple of other related newborn situations to watch for.......
One is narcalepsy (sp) where the foal will sleep so much that it won't wake up to nurse! We have had two foals like this and both out grew it in a couple of weeks. We did, however, have to watch them 24/7 and make sure they nursed ever 2 hours.
The other problem is a RH Factor foal. If a foal acts listless or falls asleep every time it nurses - similar to both above situation, you need to get mom and foal to the vet ASAP. In essence the colostrum is killing the foal and you will have to find another colostrum source........Then the mare's colostrom needs to be milked from her. That can take up to 36 hours. Once the colostrum is no longer being produced and just milk is coming in, the foal can go back to nursing off its mom.
Don't mean to scare you, but sometimes a "fainting foal" can mean other things as well and the differences can be subtle.
MA