Is it permissible to burn chametz that contains shemitah produce?

Answer

On the one hand, the chametz prohibition obligates us to destroy and get rid of any chametz in our ownership. On the other hand, there is an obligation to eat shemitah produce – "לאכלה ולא להספד ";  that shemitah produce is for us "to eat and not to waste." This constitutes the biblical  hefsed prohibition. If we have chametz prepared with shemitah produce (such as banana bread, cookies with shemitah wine, or olive oil, etc.) what are we supposed to do?

Rabbi Netanel Oyerbach wrote a comprehensive article on the topic in the periodical Emunat Itecha, issue 111, Nissan 5776 (the first year of the shemitah cycle), pp. 52-59: "Burning chametz with shemitah produce on erev Pesach" (Heb.). See the full article on our website, in PDF here and a summary of the article was posted on Arutz 7 here. 

Rabbi Oyerbach approaches the topic from the angle of burning terumah on erev Pesach since terumah is equated to shemitah produce for many halachot. The gemara (Bavli, Pesachim) notes that one is obligated to burn terumah chametz. Rabbi Unterman disagrees, contending that shemitah produce differs from terumah as shemitah has an independent injunction of "to eat and not to waste." For this reason, he maintains, it would be forbidden to burn shemitah chametz as the Torah prohibits hefsed.

Rabbi Oyerbach proves otherwise, that the injunction of "to eat and not to waste" is part of the laws of eating shemitah produce. Therefore, when it is forbidden to eat this produce, the laws of hefsed no longer apply. His proof is from the Rambam, who at the bi'ur time for shemitah produce (not bi'ur chametz; this is a separate mitzvah) obligates burning the shemitah produce and is not concerned with the prohibition of "to eat and not to waste." This is because at the time of bi'ur the produce is forbidden to eat, so it is no longer forbidden to waste or destroy it.

In any case, when the fifth halachic hour arrives on erev Pesach, that is, sof zeman achilat chametz, these products will be forbidden to eat. At this point we have exactly one hour when it is permitted for us to burn the shemitah chametz; now the hefsed prohibition does not apply since the food cannot be eaten.

In conclusion:

If you have chametz mixed in with shemitah produce (fruit, vegetables, wine, oil), burn the chametz at the fifth hour, when it is no longer permissible to eat chametz and before the sixth hour, when it is no longer to benefit from chametz. This is because during this window of time the chametz is forbidden to eat and burning it does not constitute hefsed.