What should one do with Shemittah produce he owns, outside of Israel?
They must be treated with Kedushas Shevi'is. That means you cannot sell them, you cannot destroy them, you cannot eat them in unusual ways, there is a Mitzva to eat them if they are kosher, and you must not continue to own them past the time they can still be found in the fields or Israel. If they grew in a field which was worked illegally by its owner during Shemitta, there is a dipute whether they may be eaten or not. See Dose of Halacha:
As one isn’t allowed to destroy shemitta produce, one can’t simply throw one’s leftover shemitta food away. Rather, one must place it in a designated place until it rots and becomes inedible before disposing of it.
Some poskim (Rabbeinu Tam, Tosafos, Sukka 39b) forbid one to eat produce that has been guarded and worked on (shamur vene’evad), while others allow one to benefit from such produce (See Igros Moshe OC 1:186).
Modern day poskim thus advise that either one eats such produce and disposes carefully of any leftovers, letting them rot first, or disposing of them all in such a way without partaking of them.