Gemara

(R. Avahu): "V'Chagarta Osam Avnet... v'Haysah Lahem Kehunah" - they are Kohanim only when Bigdei Kehunah are on them.

Hora'ah 12b (Tana d'Vei R. Yishmael): "You will be Mekadesh him (a Kohen)" - in every matter of Kedushah, to read the Torah first, to bless (after eating) first, to take a nice portion first.

Kidushin 21b (Beraisa - Chachamim): A Kohen cannot become a Nirtza, for this would blemish him.

(Rabah bar Rav Shilo): We learn from "he will return to his family" - to the status quo of his family. (Kohanim serve in the Mikdash.)

Gitin 59b (Beraisa): The one who blessed on the bread is the first to take from the bowl (to accompany his bread). He may honor his Rebbi or one greater than himself by letting them take first.

(Rabah): This applies only to a meal. Regarding reading the Torah, a Kohen may not honor another by letting him read first, lest this lead to quarrels.

Yevamos 88b (Beraisa): "V'Kidashto" - b'Al Korcho (against his will). If he married a woman forbidden to him and did not want to leave, Dafno (force him).

Rishonim

Rambam (Hilchos Isurei Bi'ah 20:13): If someone comes nowadays and says 'I am a Kohen', he is not believed to read first in the Torah or Duchan until one witness testifies about him. He forbids himself to a divorcee and to become Tamei Mes, and if he transgresses either he is lashed.

Mordechai (Gitin 461): A case occurred in which a Kohen poured water on R. Tam. A Talmid asked that the Yerushalmi says that one who used a Kohen was Mo'el! R. Tam answered that they do not have Kedushah nowadays. Their Kedushah is only when their Bigdei Kehunah are on them. The Talmid asked, if so, we should not honor them at all nowadays! R. Tam was silent.

Question (Shirei Berachah on Birkei Yosef 4, citing Yad Aharon): The Yerushalmi, which says that using a Kohen is Me'ilah, discusses nowadays!

Nish'al David (OC 4 DH Ach): The Yerushalmi learned from a Hekesh of Kohanim to Kelim. Nowadays, the Kelim are not Kodesh - "u'Va'u Vah Paritzim v'Chileluha." Also Kohanim may be used. We do not learn about other Kedushos of Kohanim, so R. Tam did not need to answer the Talmid.

Rivash (94): We need not punish greatly one who disgraces a Kohen Am ha'Aretz, since the Kohen does not learn Torah like a Kohen should. All the more so this applies nowadays, when Kohanim cannot prove their lineage.

Maharshdam (EH 235): The Rivash says that Kohanim today do not have Vadai lineage. We rely on Chazakah to let them read first in the Torah, which a mere custom. The Isur of a captured woman to a Kohen is only mid'Rabanan. It suffices to forbid to Vadai Kohanim. It is a Sefek-Sefeka (two doubts), for even if he is a Kohen, perhaps she was not defiled.

Shevus Yakov (1:93): The Shulchan Aruch (EH 6:1) forbids a Kohen to a Vadai Chalutzah, even though the Isur is only mid'Rabanan. If a Kohen married someone Pasul to him, we excommunicate him and apply other stringencies until he divorces her. The Rivash (348) says that we force one to divorce a spouse forbidden mid'Rabanan, and all the more so for Isurei Kehunah. The Maharit (149) agrees that nowadays we force only for a divorcee. To avoid a contradiction with the Shulchan Aruch and Rivash (348), we must say that the latter discuss Kohanim of Vadai lineage, if any are found nowadays. However, the Maharit and Chut ha'Shani (17) say that Kohanim nowadays are not considered Safek. They learn from 'Chazakah is great.' Tosfos explains that the Gemara discussed when there was a flaw in their Chazakah, i.e. they could not find their lineage documents. The Maharit same applies to Kohanim nowadays; they are only Safek. The Gemara concluded that the Kohanim who returned with Ezra and could not prove their lineage could eat only Terumah mid'Rabanan, lest people assume that their lineage is Vadai. I disagree. The Gemara concluded that the Kohanim who could not prove their lineage could not eat any Terumah! Yevamos 99a discusses stringencies of a Safek Kohen. It does not say that he cannot Duchan! Rather, even though a Zar may not Duchan, a Safek Kohen may, for a Kohen who does not Duchan transgresses three Mitzvos Aseh. Since mid'Rabanan he may Duchan, he may bless. The Maharit says that even if he is not Muchzak regarding others, Shavya a'Nafshei Chatichah d'Isura (by saying that he is a Kohen, he forbade himself to all Isurei Kehunah). It seems that this is only for Isurim mid'Oraisa. The Rambam and Tur say that he forbids himself to a divorcee, Zonah and Chalalah. They omitted Chalutzah. Really, Shavya a'Nafshei forbids even Isurim mid'Rabanan (Tosfos Sanhedrin 54b DH Omar) However, here the Kohen himself is unsure. In summary, it is great if he agrees to divorce her, and we try to verbally coerce him, but we do not force him physically or through Cherem.

Be'er Heitev (EH 6:2, citing Keneses Yechezkeil 56): Chalilah to denounce the lineage of Kohanim nowadays! If so, no one will redeem firstborns!

Mishneh l'Melech (Hilchos Kinyan 3:8): Yefeh Mar'eh (Yerushalmi Berachos 8:2) learns from the Rivash that one may use a Kohen of uncertain lineage. This is no proof. The Rivash said that we do not punish so severely for disgracing a Kohen, for his lineage is uncertain. However, one may not use even a Safek Kohen! If one established himself to be a Kohen, if he marries a divorcee we lash him, even without proof of his lineage.

Chasam Sofer (Kovetz Teshuvos 58 DH Hevi): The Rivash merely said that we do not fine for disgracing a Kohen, for he can say 'prove that he is really a Kohen.' One may not rely on this to be lenient

Poskim

Rema (OC 128:45): One may not use a Kohen even nowadays, for this is like Me'ilah with Hekdesh, unless he pardoned his honor.

Taz (39): The Kohen who served R. Tam was happy that he merited to serve Kodesh. This is his Kedushah! R. Tam was not silent because he could not answer. Rather, he did not want to say 'it is permitted, for I am a Chacham.'

Beis Yosef (YD 251:9 DH Kosav ha'Rambam): The Yerushalmi says that a Levi has precedence over a Yisrael only at the time of the Duchan (when Leviyim sing in the Mikdash), but nowadays there is no precedence. The Bavli and Poskim do not make this distinction.

NOTE: It seems that if the Beis Yosef held that some hold that Kedushas Kohen does not apply nowadays, he would have discussed this. Rather, he explains like the Taz, that R. Tam did not want to reveal the real Heter. (It could be another reason, e.g. because the Kohen was of uncertain lineage).

Mishnah Berurah (174): Even nowadays he is Kodesh. This is why he may not marry a divorcee or become Tamei Mes.

Magen Avraham (201:4): Why are we lenient about Kedushas Kohen nowadays? Perhaps it is because their lineage is not certain.

Rebuttal (Be'er Heitev 128:83, citing Keneses Yechezkeil): Heaven forbid to cast aspersion on the lineage of Kohanim nowadays!

Minchas Chinuch (269 DH v'Ayen Sham and DH v'Noheges): The case with R. Tam proves that v'Kidashto applies even nowadays that lineage of Kohanim is not certain. We must be stringent about a Safek mid'Oraisa. Perhaps a Kohen must offer another Kohen to take first, for perhaps only the other is a true Kohen. Or, perhaps we may be lenient, for it is a Sefek-Sefeka.

Mayim Amukim (1:3): A Kohen and his wife were forced by a royal decree to renounce Torah, but in private they observed as much as possible. Their sons born during that period have all the Kedushah of Kohanim nowadays. We do not forbid a woman secluded with Nochrim, unless she was in mortal danger (lest she seduced her captor to try to save herself). Even though she was unable to immerse, Bi'ah Nidah does not make her a Zonah.