Mitzva Ha’ba’ah B’aveira
The פסוק says, "חג הסוכות תעשה לך," we should make a סוכה that is לך, 'for you'. The גמרא learns from this word that one is not יוצא the מצוה with a stolen סוכה, because it is not שלך, it is not yours.
תוספות points out that we have a general rule that one is not יוצא a מצוה with מצוה הבאה בעבירה, a מצוה that he only accomplished by doing an עבירה. If so, תוספות asks, of course a stolen סוכה is פסול, because he can only sit in the סוכה because of the עבירה of stealing. Why the תורה needs to write a special פסוק to teach us this הלכה?
The מנחת חינוך answers by pointing out an important difference between two types of מצות. One category is a מצוה חיובית, a מצוה that you must do, such as putting on תפילין. In such a case, you do either a מצוה or an עבירה. The מצוה is to perform a certain action, and one commits an עבירה by not doing that action. If he puts on תפילין, he gets a מצוה. If he doesn't put on תפילין, it is considered an עבירה. On the other hand, there is another category of מצות called a מצוה קיומית. You don't have to do such a מצוה, but if you go ahead and do it, you will get שכר. For example, one is not obligated to wear ציצית. Only a four-cornered garment has to have ציצית. Therefore, one can simply avoid wearing a four-cornered garment, and even though he will not fulfill the מצוה of ציצית, he will not be doing an עבירה, either.
On the first night of סוכות, we learn from a דרשה that there is a מצוה חיובית to eat in the סוכה. This has nothing to do with the עבירה of eating out of the סוכה. Even if the person didn't eat anything at all that night, he did the עבירה of not fulfilling the מצוה to eat in the סוכה. However, for the rest of the Yom-Tov, there is only a מצוה קיומית to eat in the סוכה. This means that there is no obligation to eat in the סוכה. However, there תורה does say that it is אסור to eat outside of the סוכה. Therefore, if you decide not to fulfill the מצוה, and you don't eat at all, you didn't do an עבירה, either, but it is still an עבירה to eat out of the סוכה.
Based on this חילוק, the מנחת חינוך answers the question of תוספות. It is a מצוה הבאה בעבירה to sit in a stolen סוכה. This means that the person did not fulfill the מצוה, but it doesn't mean that the סוכה is not considered a סוכה! Therefore, even though he doesn't get credit for eating in the סוכה, he also did not do the עבירה of eating outside of the סוכה, because he actually ate in a סוכה.
This means that there is a difference whether one eats in a stolen סוכה on the first night of סוכות, or during the rest of Yom-Tov. There is a מצוה to eat in the סוכה on the first night. If one eats in a stolen סוכה, he does not fulfill the מצוה, because of the idea of מצוה הבאה בעבירה. Since this a מצוה חיובית, he also did an עבירה, because he was supposed to get this particular מצוה. However, if one eats in a stolen סוכה during the rest of Yom-Tov, he doesn't fulfill a מצוה, but he also doesn't get an עבירה, because he wasn't obligated to eat in a סוכה. True, the idea of מצוה הבאה בעבירה tells us that he did not fulfill a מצוה, but after all, he still ate in a סוכה, and the only הלכה during the rest of Yom-Tov is to avoid eating out of the סוכה.
This is why we need the extra פסוק of חג הסוכות תעשה לך. This פסוק is not talking about fulfilling the מצוה. It's referring to the סוכה itself, and it teaches us that if the סוכה is not yours, it is not considered that you ate in a סוכה. If so, then it would be an עבירה to eat in such a סוכה, even during the other days of Yom-Tov, because you are never allowed to eat out of the סוכה.
