The Torah says that the lulav used for the mitzvah must be owned, "lachem," and therefore a stolen or borrowed lulav is disqualified on the first day of Yom Tov. After the first day, when the mitzvah is only derabbanan, even a borrowed lulav is acceptable. The gemara at the beginning of the third perek of Sukkah quotes a machlokes between R' Yochanan and Shmuel regarding whether a stolen lulav is usable on the remaining days of Yom Tov. R' Yochanan writes that a stolen lulav is disqualified for the duration of Yom Tov because of mitzvah haba'ah b'aveira. Shmuel argues that since a borrowed lulav is acceptable on the remaining days, a stolen lulav is also acceptable. Tosfos asks, a lulav or esrog which is disqualified because it is not hadar is unacceptable for all eight days of Yom Tov -- why does the disqualification of not being hadar apply for the duration of Yom Tov but not the disqualification of being stolen? How does the fact that a borrowed lulav may be used justify using a stolen lulav?
Perhaps one could answer the question by using the chakirah of the Brisker Rav which we discussed in the past -- is the takanah of taking lulav for the duration of Yom Tov done zecher l'mikdash an extension of the mitzvah of lulav or a new chiyuv of making a zecher l'mikdash? If it is a new chiyuv of zecher l'mikdash, one can perhaps distinguish between the psul of hadar, which is inherent and visible in the object of lulav and esrog, and the psul of gazul, which relates to the behavior of the person performing the mitzvah but has no visible effect on the object.
Since I mentioned my wife's grandfather's Telzer roots, I can't resist offering a Telzer answer. R' Yosef Leib Bloch explains that the machlokes R' Yochanan and Shmuel revolves around how to define mitzvah haba'ah b'aveira. The Sha'ar haMelech suggests that mitzvah haba'ah b'aveira applies only where the performance of a mitzvah could not be accomplished if not for the aveira. Based on this, the gemara's explanation of Shmuel -- "m'toch she'yotzei b'shaul yotzei b'gazul" -- makes perfect since. The very possibility of borrowing a lulav disconnects the act of theft from being an integral part of the mitzvah and disqualifying it.
R' Yosef Engel (Lekach Tov 12:2) takes a more pilpulish approach. He notes that Rashi on the mishna explains the requirement of hadar based on the pasuk of "zeh K-li v'anveihu", which is the source for hidur mitzvah with respect to all mitzvos. Why would Rashi not cite the pasuk of "pri eitz hadar" which applies specifically to the 4 minim instead of the more general din of hidur mitzvah? R"Y Engel explains that the psul of "hadar" written by the 4 minim is me'akeiv even bdi'eved; the general rule of hidur mitzvah is not. What disqualification the Mishna is referring to underlies the dispute between R' Yochanan and Shmuel. The Mishna lumps together the disqualifications of a stolen lulav and a lulav that is not hadar. If the Mishna is referring to the general din of hidur mitzvah, than just as it is not me'akeiv, so too the psul of gazul is not me'akeiv as a disqualification on other days except for the first. If the Mishna is referring to the specific din of hadar, then just as it is me'akeiv throughout sukkos, so too, the din of not taking a stolen lulav is me'akeiv.