Because the shofar is an instrument of melacha, you cannot carry it on Shabbes regardless of whether there is an eruv.
Melacha? What melacha is performed by blowing air into a horn? I don't see how that is a melacha but maybe I'm wrong. Also in the quote muman posted, they did not say it is an instrument of melacha, in fact rabba said we should blow shofar even on Shabbath. And then, indeed, the article says that the reason we don't is that in the end, the issue of carrying was the problem.
"The Gemara (Rosh Ha-shana 29b) cites Rabba's assertion that as far as Torah law is concerned, we should blow the shofar on Rosh Ha-shana even if it occurs on Shabbat. The Sages, however, decreed that the shofar not be blown on Shabbat, as one may neglectfully carry his shofar through the public domain (which is forbidden on Shabbat) in order to practice blowing under the tutelage of an expert."