When I did it, I only ever had one heart to work with at a time so I did refrigerate it. The recipe that I adapted mine from did not specify but did state that it will keep for one year.
Pickled Heart
1 heart (game or beef heart)
1 tsp salt
1 tbsp pickling spice
1/4 onion
sugar
vinegar
water
Rinse the heart in fresh cool water, poking your finger into the
ventricles to help remove any blood clots. Soak overnight in a cool
place with enough water to cover plus a handful of...
I have a neighbor that gave me a beef heart so I pickled it. Once she got a taste of the pickled heart, she never gave me another heart. I also like heart stuffed and roasted.