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.
Parking fees rise at two Devon attractions
Increase of 10 percent
Street hubs planned for Exeter
'They are a sleek and modern answer'
Sex abuse masseur jailed for four years
Ian Nicholls, 64, assaulted clients
'Safeguards-on-asylum-seekers' Reform council voted down
Even his seconder had second thoughts
Devon care home fined £200,000 after two residents drowned
Burdon Grange failed to provide safe care
Residents urged to help rough sleepers
Volunteers can join the soup run
