In order to answer this question, we must first evaluate whether it is permitted to blow the shofar after the mitzva is completed. If it is permitted, then the shofar is a kli shemelachto l’heter (a utensil used for permitted purposes) and can be moved for any reasonable purpose.
This matter is actually the subject of substantial disagreements. The Rama[1] says that one is forbidden to blow the shofar for no specific need on Rosh Hashana after he has fulfilled the mitzva.[2] The Tur[3] even cites an opinion that a man may not blow shofar on Rosh Hashana for a woman after fulfilling the mitzva himself, since she is not obligated to hear shofar blowing, and it is therefore a blowing without justification. This opinion is not accepted because we rule that there is nevertheless value for a woman to hear the shofar. Otherwise, though, it would have been forbidden.
Another discussion[4] mentions an opinion that one who makes the shofar blower repeat a blast when he is not required to do so is making him violate the Rabbinic prohibition of blowing the shofar without a reason. The gemara[5] also discusses the case of children blowing the shofar for practice for the future without a mitzva need and implies that an adult should not do this.
On the other hand, the Taz[6] argues that blowing shofar is forbidden only on Shabbat, out of concern that one may carry the shofar in the public domain, but it is permitted on Yom Tov, since carrying is permitted then. The Taz maintains that the sources forbidding blowing for adults refer to Shabbat. The Ohr Zarua,[7] in fact, explicitly permits blowing shofar all day long on Rosh Hashana that does not fall on Shabbat.
In the final analysis, most poskim forbid blowing shofar without a specific purpose after fulfilling the mitzva. However, one may blow to fulfill the minhagim to add more blasts (100, for most) than are strictly required for the mitzva.
One of the Taz’s arguments that it is permitted to blow after finishing the mitzva is that if it were forbidden, then it would also be forbidden to move the shofar due to muktzeh. We would similarly expect that it would be forbidden to carry a lulav after fulfilling that mitzva. However, the gemara[8] assumes that it is permitted to carry a lulav after fulfilling the mitzva, and the Rama[9] himself says that a shofar is muktzeh only regarding Shabbat. The Magen Avraham[10] contends that a special type of muktzeh applies to a shofar, namely, muktzeh l’mitzvato.[11] This form of muktzeh primarily precludes using the shofar for another purpose, but many assume that muktzeh l’mitzvato does not prevent one from moving the object.[12]
In any case, the great majority of poskim assume that even though one may not blow the shofar beyond the needs of the day, he may move the shofar.[13] The Mishna Berura[14] explains that one cannot presume that a shofar is no longer slated to be used, because it is possible that someone has not heard shofar-blowing and still needs it. The Kaf HaChayim[15] reasons that since children may blow throughout the day, the shofar is fit for use, and thus not muktzeh.
The Shemirat Shabbat K’Hilchata[16] wonders why we assume that one may move a shofar after having finished using it, while it is unclear whether one may move a brit mila knife after the mila.[17] He suggests a logical distinction – it is more common to unexpectedly find someone in need of hearing shofar-blowing than to unexpectedly find a baby in his eighth day who needs a mila.