If there are a pair of brothers (Reuven and Shimon) who’s father is a kohen and Reuven decided to forego his kohen status to marry someone, is Reuven still called to the Torah as Reuven ben Ploni haKohen, or just Reuven ben Ploni? (Shimon is in the same kehillah and is called up as the kohen he identifies as).
First of all – there IS NO SUCH THING as giving up your kohen status. You can violate your kohen status, but you are still a sinning kohen, not a non-kohen. Now, that being said, when a kohen marries a woman forbidden to him, there is an element of compromising his status, since the children of that union – but not him, himself – are chalalim, and even that aside, by violating a prohibition that is unique to his kohen status, it is considered (rabbinically) as if he violated / desacralized this status. He is still bound by all the laws, but the community, in practice, affords him none of the honored status of a kohen – he does not do birkat kohanim or get the first aliyah, etc. See Shulchan Arukh OC 128:40,
כהן שנשא גרושה, לא ישא כפיו ואין נוהגין בו קדושה אפילו לקרות בתורה ראשון
And Mishnah Berurah 128:147:
ומה שנתבאר בסמוך שאין שאר עבירות מונעים נ”כ היינו דוקא בעבירות שאין שאר הכהנים מוזהרים יותר מישראל אבל במה שהכהנים מוזהרים יותר מישראל שקדושת כהנים גרם לו והוא חיללו לפיכך פסול מדרבנן לכל דבר כהונה עד שידור ע”ד רבים וכו’
The only way to fix it is to divorce his wife and take an oath that cannot be undone that he will not remarry her or any other woman who is forbidden to him.
So – as to your question – the key question is whether הכהן refers to him or his father. This comes up in modern ketuvot – do you write – אליעזר בן אהן ושרה הכהן or אליעזר בן אהרון הכהן ושרה. I should say that I was initially shocked when I saw that Rambam signed his name as משה בן מיימון ז”ל, until I realized that the z”l was going on the father. So – in our case – even if it goes on the father, without saying anything else, it suggests that he is a kosher kohen. That is really only a problem in the case of a chalal, who genuinely isn’t a kohen and has no kohen restrictions. In our case, he is a full kohen without the honors of a kohen. So you WOULD say ראובן בן פלוני הכהן, regardless, although you would only call him up for an aliyah from the third aliyah onwards (and see Iggrot Moshe OC 2:33).
The general assumption is that הכהן does refer to the father – see Rambam example, above. Please take note those who are writing ketuvot with mother and father’s name – either write אליעזר בן אהרן הכהן ושרה or use a comma – אליעזר בן אהרן ושרה, הכהן.
As stated, the real question is with a חלל. Even though הכהן can be assumed to refer to the father – not doing anything different sends the message that he is a kosher kohen. This becomes an issue in particular in ketuvot which are looked at for personal status. There are 3 opinions there: (a) do it regularly – פלוני בן פלוני הכהן – since the chalal himself is not forbidden to marry anyone (although his children will not be kohanim); (b) drop הכהן, (c) write החלל, i.e. – פלוני החלל בן אהרן הכהן. See here –