Note: See Rav Sternbach below:
There is no obligation to declare the Zero'a, Lechayayim, and Keivah as sanctified, or to separate them from the animal, if they are not going to be given to a Kohen.
This implies that if one cannot get the portions to a kohen before it spoils, then one need not separate them to begin with. If Shem was the Kohen, then Avraham could not have gotten it to him and was patur from separating it.
The gemarah Megilla daf 28a brings the case of Rav Preidah who said that as a middus chassidus he had never eaten from an animal from which the matanos had not been taken. The Art Scroll edition of the gemora on daf 28a (1) footnote 4 brings the Ritva who says that
The Gemora concludes that there is no analogy between the meat of an animal whos gifts have not been separated and tevel. ... Also in contrast to terumah, the priestly gifts from an animal are non-sacred, and so may be eaten even by a non-Kohen. Nevertheless, Rav Preida went beyond the demands of halacha, refusing to eat meat from an animal whose priestly gifts had not been separated (Ritva)
Avraham kept the Torah as a midus chassidus and on that basis is halachically allowed to eat tongue as well as serve it to his guests.
On thinking it over further, I have come to the conclusion (through logic) that Avraham would have also been considered a kohen at that time (when the "guests" arrived). In fact, that is why we have the line of kehunah being thought of as Shem (who outlived Avraham), Aiver, Avraham, Yitzchak, Yaakov, Levi (as seen by the treatment of the tribe of Levi in Mitzraim). It was only after Matan Torah that we have the assigned kehunah and the specific matanos.
The logic involves the fact that Shem (Malchitzedek) is called a kohen of Hashem. From the way Avraham is treated (Nsi Elokim) by Avimelech and Bnei Ches (for example) it would appear that he was also recognized as a kohen. At that time, the kehunah was not a matter of yichus, as that only started after the appointment of Aharon.
Since we can treat Avraham as a kohen, he was allowed to assign the matnos kehuna to himself, and, since they were not kodesh, serve them to his guests.
UPDATE The section below explains why tongue is permitted even though it was not given to the Kohen. This would apply to Avraham as well as nowadays.
INSIGHTS INTO THE DAILY DAF - Chulin 130
(c) RAV MOSHE STERNBUCH shlit'a explains that there is a basic
difference between Pidyon ha'Ben and all other Matnos Kehunah. The
obligation to redeem a firstborn son is independent of the five-Shekel
payment to the Kohen. Even when the five Shekalim are not given to a
Kohen, it is necessary to "remove the Kedushah" from a Bechor by
separating five Shekalim as a Pidyon. Since it is necessary to
separate five Shekalim regardless of whether it will be given to a
Kohen, the father must give them to a Kohen even though the Kohen is
not fully "Muchzak." In contrast, Zero'a, Lechayayim, and Keivah (and
Reishis ha'Gez) involve nothing more than a monetary obligation to the
Kohen. There is no obligation to declare the Zero'a, Lechayayim, and
Keivah as sanctified, or to separate them from the animal, if they are
not going to be given to a Kohen. Since one does not need to give them
to a Kohen who is not "Muchzak," one also does not need to separate
them from the animal.