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I have a loquat tree in my yard, which I generally cover in paper bags regular years to prevent birds from picking at them.
1) Shemitah fruits are hefker also to animals and it is forbidden to prevent them from eating them. When there is a concern that animals will destroy fruit and prevent people from eating it, it is possible to prevent their access to the fruit. For this reason, it is permissible to cover the fruit to prevent animals from eating them.
2) The Torah prohibits four agricultural activities and add "any work in the fields and vineyards" and distinguish between when failing to perform certain activities will cause loss (ukmei) and acts whose goal is enhancement of growth (ukmei peira). Rabbi Avraham Yitzchak HaKohen Kook maintains that shemitah crops are ownerless and humans are not the owners of the fruit, which is why any actions to save the crop from spoiling is forbidden. However, there are posekim who permit this, holding that the crops are not ownerless per se, rather they are not owned by any individual, rather by the entire Jewish People.
The question of if it is possible to wrap fruit and saving it depends on this dispute. According to Rabbi Kook, covering fruit in paper bags is a distinctly agricultural and is forbidden, just as he would forbid covering the tree in netting to save the fruit (another agricultural act).
However, non-agricultural activities are permitted, even if their purpose is to save the crop. For this reason, Rabbi Kook permitted trapping pests in orchards that are installed by digging in the soil, since this is a non-agricultural act. Similarly, it is permissible to install a scarecrow to scare away pests, since this, too, is a non-agricultural act.
In practice it is best to avoid engaging in agricultural activities that save crops from destruction, following the opinion of Rabbi Kook. However, it is possible to rely on the lenient opinions (Chazon Ish and Rabbi Yisraeli) for those who find this difficult.
[Note by Rabbi Moshe Bloom: Even among the rabbis at Torah VeHa'aretz Institute there is a dispute on the practical ruling for this question. The reason for this is that when one is stringent and avoids covering the fruit, often no edible fruit will remain on the trees and thus the person will not merit to eat the shemitah produce and fulfill the mitzvah of "And the produce of the land's Sabbath will be for you to eat."]