The braisa lists teruma and bikurim as things that a non-kohen is chayev misa for eating AND they are assur for a non-kohen to eat. Rashi and Tosafos are both bothered why the braisa would have to tell us that there is an issur for zarim after it already says that a zar who eats it is chayev misah. Rashi explains that it is to tell you the contrast of ma'aser, not only is there no misa for a zar, it is even permitted. However, Tosafos explains that it is to tell you the chiddush that even a partial shiur of teruma and bikurim are assur for a zar (according to r' yochanan that chatzi shiur is assur m'doraysa, it would be assur m'doraysa, and according to reish lakish it would only be d'rabonon). The Maharatz Chiyus quotes from the shita who asks, why is this a chiddush? Why should teruma and bikurim be different than any other issur in the torah where chatzi shiur is assur?
It seems that there is a chiddush in the case of teruma and bikurim. The source of chatzi shiur being assur (yoma 74a) is a combination of the pasuk of כל חלב complimented by the principal of חזי לאיצטרופי. The issur of chatzi shiur may very possibly be limited to ma'achalos asuros such as cheilev, but would not apply to teruma and bikurim which are not object of issur. Meaning, the concept of chatzi shiur may only apply to issurei torah that are a "cheftza shel issur", similar to cheilev, where we say that even a small portion is considered a cheftza shel issur (just without malkus). But teruma and bikurim are objects of mitzvah, not objects of issur. The issur on a non-kohen to eat them is an issur gavra - where one can argue that we don't apply the concept of חצי שיעור אסור מן התורה. The braisa is mechadesh that we apply the concept of chatzi shiur even to an issur gavra, just as we apply it to timebound issurim which are also issurei gavra in nature (as we see from the tzlach in pesachim 44a and repeated in his nodeh b'yehuda tinyana o.c. 53, where only at the end of the day where we no longer have חזי לאיצטרופי would we say that there isn't an issur of chatzi shiur).
To explain this a bit better, I have a discussion in my sefer as to how we apply chatzi shiur to a mitzvas aseh. The concept may not apply at all to a mitzvas aseh. The issur of a non-kohen to eat teruma and bikurim may be a result of the will of the Torah that a kohen be the one to eat teruma (although there isn't any specific mitzvah on a kohen to eat teruma, just an issur on a zar not to eat it). Since a chatzi shiur isn't considered a significant achila for a kohen to eat, it may not be an issur for a non-kohen to eat. The braisa is mechadesh that there is in fact an issur for a non-kohen to eat a chatzi shiur of teruma and bikurim.