The gemara says that the machlokes whether women are allowed to blow shofar on RH is contingent on the machlokes whether they are permitted to do semicha on a korban. The gemara is not clear what the issue would be. Rashi understands that it is a general issue of bal tosif for women to do mitzvos that they are not obligated in, whereas the Ran in Kiddushin discusses it as more specific to these mitzvos (semicha is an issue of avoda b'kodshim, and shofar is an issur d'rabonon as Tosafos writes on 29b).
The gemara presents the opinion of R. Yossi (which Tosafos holds we pasken like) to say that it is a "reshus", which implies that there is nothing wrong with them doing it, but perhaps there is no advantage either. Tosafos then develops that according to R. Yossi it isn't merely a d'var ha'reshus, but it is considered a mitzvah and enough of a mitzvah for them to even make a bracha on it. Tosafos elaborates with ra'ayos to prove that women can even make a bracha on these mitzvos.
R. Akiva Eiger in the Gilyon HaShas cites the Hagahos Ashri at the end of the third perek of Succah who disagrees. The gemara/rashi in Succah implies that a lulav and esrog is not muktzah for a woman to move because since it is needed by a man, it has the status of a keli. The Hagahos Ashri deduces from this that women would not be able to make a bracha on the lulav and esrog, otherwise the gemara should have said that it isn't muktzah for them simply because they fulfill a mitzvah with it. Although the focus of the Hagahos Ashri is to argue on Rabbeinu Tam who permits a bracha to be made, it seems clear from the proof that he cites that there isn't any aspect of mitzvah that is achieved when a woman takes lulav and esrog. If there would be any aspect of a mitzvah performed in their taking lulav and esrog, it would certainly not be muktzah even if there were not men in the world who would be obligated. Therefore, it seems that the Hagahos Ashri is two steps removed from Rabbeinu Tam, by rejecting the right to make a bracha, and the considering it to be a mitzvah.
The Rambam (Hil. Tzitzis 3:9) writes that women are exempt from tzitizs but...
נשים ועבדים שרצו להתעטף בציצית מתעטפים בלא ברכה, וכן שאר מצות עשה שהנשים פטורות מהן, אם רצו לעשות אותן בלא ברכה אין ממחין בידו
The Rambam seems to hold like the Hagahos Ashri. But, why would women and avadim "want" to do these mitzvos? Is it just a mishe'gas with no value!? It seems that the Rambam would hold that there is a mitzvah for women to perform these mitzvos, and they receive reward like an אינו מצווה ועושה, but are not allowed to make a bracha. This is how the M.B. in Hilchos Shofar (589:6) explains the position of the mechaber who says that women are patur but are allowed to blow shofar on Rosh Hashana. Even though it is considered a bit of an issur to sound the shofar, they are allowed to do it since it is considered a mitzvah for them for which they receive reward. Its well known that the Rama rules in Hilchos Tzitzis, Tefillin, Shofar and Succah like Rabbeinu Tam, but even the mechaber who rules like the Rambam would seem to consider it a mitzvah, unlike the Hagahos Ashri.