The following side effects have been reported: allergic reactions, tbloating, constipation, diarrhea, gas (flatulence).
In addition to the list above, it is thought that flaxseed could interfere with the absorption of iron in the gut. This could cause a horse to become anemic, and would require a veterinarian's care.
It's also good to consider that flaxseed has not been tested for safety or effectiveness on pregnant or lactating mares so might be better to avoid it in this case to prevent any possible complications.