Tosfos (R"H 16b) asks why blowing shofar 2 times on R"H (tekiyos d'meuyshav and tekiyos in chazaras hashatz) is not a violation of bal tosif. Tosfos answers based on the halacha (R"H 28b) that there is no bal tosif if a mitzvah is performed at the wrong time, e.g. sitting in a sukkah in July would not be bal tosif. Since the mitzvah of blowing shofar is complete after the first round of blowing, the additional blasts are outside the chiyuv's timeframe -- it is like blowing shofar in July, and hence there is no bal tosif.
Tosfos rejects this answer because the halacha is that a kohein who duchens at one minyan may duchen again if he comes to a second minyan that is up to birchas kohanim -- the time of chiyuv is ongoing and does not end just because the mitzvah was fulfilled once. Here too, since during the day of R"H one can go from tzibur to tzibur blowing shofar multiple times, the entire day is considered the proper timeframe of the chiyuv.
Shu"T Chasam Sofer (O.C. 22) quotes a question from his father-in-law R' Akiva Eiger: Tosfos is comparing apples to oranges. The kohein who enters a minyan that has not heard birchas kohanim has a personal obligation to bless that tzibur; the fact that he has already blessed a different tzibur has nothing to do with this new mitzvah. However, once the ba'al tokeya blows shofar once during the day, his personal obligation has been fulfilled -- coming to a new tzibur does not in any way create a new obligation of tekiyas shofar for the ba'al tokeya.
The answer to R' Akiva Eiger's question is rooted in the relationship between birchas kohanim and tefilah. Chasam Sofer writes that birchas kohanim is inseparable from avodah/tefilah and can only be fulfilled in that context. Just as Aharon offered the original birchas kohanim after avodah, a kohein can only duchen in the context of fulfilling his obligation of tefilah. Therefore, a kohein who has already davened and comes to another minyan does not really have a new chiyuv of birchas kohanim because unlike the tzibur, he has already fulfilled his chiyuv of tefilah. Nontheless, as Tosfos writes, the gemara does not consider his repeating birchas kohanim to be a violation of bal tosif, and the same logic might apply to tekiyas shofar.