I think they would have to eat a fair amount to be put at risk,
Poisoning by oak is usually seasonal, being most common in spring when the young buds or leaves are eaten and the autumn when the acorns are eaten. Oak leaves and acorns contain tannic acid which is poisonous to horses and though eating a small number of leaves or acorns is almost certainly harmless, they can also be addictive, and once a horse has acquired a taste for them they can actively search them out. Also some animals seem to be more suseptible to oak poisoning than others with individual animals having different levels of tolerance.
Oak poisoning causes gastroenteritis and kidney damage.
Symptoms include:
lack of appetite
staring coat
constipation followed by diarrhoea which may be bloodstained
abdominal pain
depression
blood in urine
There is no antidote. The horse is treated with drugs to reduce the pain and control the diarrhoea, antibiotics may be prescribed.
Prevention
In general it is best to restrict the access of horses to acorns, particularly if other food is scarce,or else pick up the fallen acorns at least once a day - although this method is time-consuming and less effective as most horses will still find some. The best thing to do is fence off oak trees - either permanently or with electric fencing.