If someone comes these days and tells us he's a kohen, he is not believed and we don't elevate him to be a kohen according to himself, and he does not get the first aliya and does not duchen. Hagah: Some say he is believed to get the first aliya and to duchen these days since we don't have terumah dioraissa that we worry [giving him the first aliya and letting him duchen will cause people to give him] terumah (it is a capital offense for a non-Kohen to eat terumah), and this is our practice these days; there is nothing to worry about.
He must not eat kodshei hagvul (i.e., terumah) until he has one witness [testifying that he's a kohen] but he prohibits himself fro mmaqrrying a divorcee, zonah and chalallah; he may not become tamei from a corpse, and if he married [one of these women] or became tamei he receives lashes...
A compendium of the nosei keilim on this halacha:
Why we shouldn't give him Kohen aliya-
Not much of a problem nowadays that there is no real terumah. (I guess we must assume even in Eretz Yisrael bizman hazeh truah is dirabanan which is fine according to most poskim.) If there would be Terumah, even Terumah dirabanan, we wouldn't give him kohein because if people see him getting the aliya, they will give him Terumah. It's not a real problem to give a non-Kohen the kohein aliya - it's only a darkei shalom issue to give a kohen the first aliya (see Chelkas Mechokek E"H 2:1). Beir Heitev asks why aren't we worried someone will redeem their firstborn son through him and he will be over the isur of stealing, and he does not have an answer. (As an interesting aside, I saw at one place they'd give the kohanim teruah oil with which to light their chanukiot.)
Why he can't duchen-
- It's an isur asei (ko tevarchu - atem velo zarim)
- He can't make the bracha (AKBV levareich as amo yisrael b'ahava, abbrv. AK)
- Only in Mikdash is a problem because the Birchas kohanim there included the shem hameforash (see Kiddushin 71a) but bizman hazeh that's not an issue
- Aruch hashulchan O"C 128:7 if there are other kohanim and he doesn't say the bracha AK, he may say the words of yevarechecha etc. because they are only psukim. But if he's the only one up there, it's asur.
A known Kohein who married a grusha and claimed he didn't know that he really was a kohen - we force him to divorce her (Levush, quoted at end of Beis Shuel 7). Beir Heteiv 6 adds the firstborns who were dedeemed by him do not need to re-redeem themselves with another kohen.
If he's not really a kohen and he redeems a bechor, he is stealing! Rav Yaakov Emden therefore says he should return the money afterward: if he's really not a kohen, it's not his, so he's returning it. If he really is a kohen, even though it's really his, it's a gift back. This is because many hold kohanim bizman hazeh are only muchzakim (as opposed to meyuchasim). There is a
faomus story of the Gra - he found a member of the Rappaport family who were meyuchas as kohanim and re-redeemed his bechor from that Kohen.
There is an important teshuva of R' Moshe Feinstein, see Shut Taam Baruch by R' Baruch Leizerovsky E"H 70 (Yay! I just found it in Igros Moshe Even Haezer 4 siman 39 using Bar Ilan Responsa Online) says that if the only evidence that someone is a kohen is from irreligious people, we say he's not a kohen. (If he has religious relatives who are kohanim, we can't say that he's not a kohen.) And he can then marry a woman who would be forbidden to him as a kohen, and he may become tamei meis. I heard Rabbi Gedalia Dov Schwartz mention situations where he has used this. Some poskim, however, have much more trust that someone who was a kohen really is a kohen and would be unwilling to rely on Rav Moshe's psak.