The Gemara speaks about teruma and matnot kehuna. It mentions Tannaim who lived generations after the churban bayit as giving or receiving teruma from the granot.
(a) Obviously teruma and matnot kehuna were given after the churban. Were kohanim able to maintain tahara? There couldn't possibly be a para aduma sprinkling afterwards could there be?
(b) When did it stop? Why did it stop? Was it related to chazaka in kehuna which we don't have any more? Was it Roman prosecutions? Were they worse 300 or 400 years after the churban than 200 years after? Were any korbonot still brought after the churban?
Matnos Kehunah are given even today: Pidyon ha'Ben is Matnos Kehunah. As for Taharah, Terumah of Chutz la'Aretz may be eaten even by a Tamei Mes. It is Asur only for one who is Tamei a "Tumah ha'Yotzei mi'Gufo", which means that a Mikvah suffices. (See Bechoros 27a, Rambam Hilchos Terumos 7,8).
Whether or not our Chazakah is valid is a Machlokes Achronim; see Pischei Teshuvah YD 305:12 and Teshuvah of the Chasam Sofer printed at the end of Chidushei Kesuvos.
Dov Zupnik