The Mishna in Sukkah (דף כט:) says לולב של אשרה ושל עיר הנדחת פסול. This means that a Lulav from a tree which was used for עבודה זרה, or from a city which is to be destroyed (because its inhabitants worshiped ע"ז) is invalid. The Gemara in Sukkah ( דף לא.) says that אשרה (a tree used for ע"ז) refers to אשרה דמשה (an אשרה from the time immediately following Moshe's death). Upon entering Eretz Yisrael from the desert, the Jews were commanded to conquer the land. At the same time they had an additional responsibility of burning all of the אשרה trees in the land. The Gemara says, that since the אשרה דמשה has a requirement of being burned, it is considered to be כתותי מכתת שיעורי, a tree that no longer meets the minimum measurement because it is about to be burnt; it is therefore פסול.
Rashi (דף כ"ט: ד"ה ושל עיר הנדחת) explains that a Lulav which has a חיוב to be burnt is כשרוף דמי, as if it has already been burnt. One may not use this Lulav, since it is כשרוף דמי, and he is merely holding ashes, which bear no resemblance to an actual Lulav. At the same time, however, there is a contradiction because Rashi gives another reason that one may not fulfill the Mitzvah of Lulav with a Lulav which must be burnt, namely, that this Lulav fails to meet the minimum required measurement. This last statement of Rashi implies that had there not been a minimum size requirement for Lulavim, one could be Yotzai with a לולב העומד לשרפה. Rashi, however, just said that a לולב העומד לשרפה, is כשרוף דמי, so it is no longer a Lulav at all. Therefore, the problem with using לולב העומד לשרפה is that it is mere ashes and not a Lulav. The minimum size requirement is irrelevant.
The Gemara in Chullin (דף פ"ט.) makes the question even stronger. The Gemara says that everything in an עיר הנדחת must be destroyed. Therefore, a Shofar found in the city is פסול since it lacks a שיעור. The Gemara continues to say that one may use the dirt from an עיר הנדחת for כיסוי הדם, covering the blood of a slaughtered bird or wild animal. The reason for this is that there is no minimum amount of dirt required for כיסוי הדם. The question remains, why does the concept of כשרוף דמי only apply to an object which has a minimum size requirement?
The answer to this dilemma may be found in a deeper examination of the meaning of the שיעור. When one talks about something needing a שיעור, this really means that it needs importance.
One example of this concept can be found by the הלכה of בריה. This means that one who eats an ant violates the prohibition against eating insects, despite the fact that the ant is less than the minimum requirement for most dietary prohibitions. The reason for this is the ant has importance since it consists of a whole unit. Accordingly, it is possible to have a שיעור even when the object lacks the requisite size.
Another case where the whole purpose of the minimum shiur is to define importance is by הלכות שבת. One must carry the amount of a dried fig to violate the prohibition of carrying on Shabbat. However, the Mishna in Shabbat (פרק י משנה א) tells us that if someone sets an object aside as being special before Shabbat, he may not carry that object even if it is smaller than the size of a dried fig. Anyone else, however, can (on a דאורייתא level) carry that same object if it is smaller than a dried fig. This is a clear example of how the minimum size requirement is there only to define importance. Thus, an object lacking the minimum size will still be included in this prohibition if it has importance. Conversely, we can now say that something of proper size that is essentially worthless has no שיעור! This now explains Rashi very clearly. When Rashi says כל העומד לשריפה כשרוף דמי he does not mean that one is holding mere ashes. Rather, this Lulav has lost its importance. Hence, despite its size, it does not meet the minimum requirement, and it can not be used as a lulav.
(This article is a summary of a shiur I heard this summer in NCSY Kollel from my Rebbe, Mark Smilowitz)