It follows the outbreak of equine flu in horses.
A message on the charity's website says while their donkeys are not affected, they're closed as a precaution.
Four more horses have tested positive for equine flu - this time in Newmarket.
Racing is on hold until at least Wednesday while the British Horseracing Authority tests horses nationwide.
Two race meetings have been cancelled at Exeter Racecourse over the past week.
Two Exeter shops hit with closure orders
Both sold illegal tobacco
Lucky Devon dog saved after complex rescue
Cockpoo falls from cliff
Plymouth up recycling with food caddies
Weekly collections coming soon
Somerset's Lando Norris wins F1 world title
He pipped Max Verstappen
City secure FA Cup Third round place
Exeter emphatically get name into hat
