I think the duplication of Emor VeAmarta, on the level of peshat, is nothing. That is, it is Biblical style, just like Daber el Bnei Yisrael, VeAmarta Aleihem. This particular construction, with Emor does not appear with great frequency, but the chataf-segol under the aleph is in place of the sheva-na, and so it is like ketov. And then to introduce the content of the speech, we have veAmarta. It is nowhere near surprising.
Shadal and Rashbam do not comment on it, and I would take there silence as meaningful -- that there really is nothing to comment on it.
Rashi does something with it, but that is in large part because Rashi engages in a hybrid of midrash and peshat. Thus:
This derasha, taken from Yevamot 114a, can be interpreted in different ways. It could just be the doubled language is a strengthening. It could be that kohanim is to refer to the adults, while bnei aharon refers to the children, the minors. I would say that Emor el Hakohanim Bnei Aharon is taken to be the instruction to Moshe, while VeAmarta Aleihem is taken to be part of the message, such that these kohanim are being instructed to say to someone else, who would be the minors. At any rate, this is midrash rather than peshat.
Speak to the kohanim: Heb. אֱמֹר וְאָמַרְתָּ “Speak [to the Kohanim …] and say [to them],” lit. “Say…and you shall say.” [This double expression comes] to admonish the adult [Kohanim to be responsible] for the minors [that they must not contaminate them (Mizrachi)]. — [Yev. . 114a]
Meanwhile, Ibn Ezra, a pashtan, has a curious suggestion, though this he presents as only one possibility:
The word taamei in the last line seems out of place, and Ramban does not record it in his citation of Ibn Ezra. I would suggest we should read it taamo, with a final vav. Not that Moshe is telling the taamei hamitzvot to them, but rather, the import of veAmarta alehem are those mitzvot only they are commanded it.
It seems like he is trying to divide the pasuk in two. Just as there was a progression from Kedoshim which was to the tzibur and parshat Emor which are commandments directed towards the kohanim, we also have the division in the pasuk itself. The kohanim are the guardians of the Torah for Klal Yisrael. Sometimes we see that to find out the law, the community goes to the kohen. So the first half of the pasuk, אֱמֹר אֶל-הַכֹּהֲנִים בְּנֵי אַהֲרֹן, just meant to say the previous parsha to the kohanim. And וְאָמַרְתָּ אֲלֵהֶם are commands directed specifically towards them, the kohanim.
Again, I don't see anything in the language justifying this, and it seems a kvetch. Perhaps he was influenced by how he could work the progression into the language.
Meshech Chochma really likes this interpretation of Ibn Ezra. He writes
what is pictured to the right, connecting it to a gemara in Sanhedin, that when a kohen offers upon the altar, they judge captial offenses. And the previous parsha was dealing with penalties for arayot, which is thus just when a kohen is sacrificing upon the altar.
Aharon ben Yosef, the Karaite scholar, also half-endorses such a peshat. He writes what is pictured to the right. That is, the repetition is because one should repeatedly tell them, so that they are warned from ritual impurity. It is surprising to entertain such a notion! But perhaps elements of this, together the next peshat, that it is merely the same as daber followed by veAmarta, is drawn from Ramban.
For Ramban critiques Ibn Ezra, and I believe that he is correct in his critique. He writes:
Thus, he rightly relegates Rashi to the realm of derash, and argues against Ibn Ezra as a matter of peshat. Rather, it is just the same as dabbeir! He also suggests the reason for this form in general, including that of daber ... veamarta is to make the warning stronger -- והוא נאמר בפרשיות אשר ירצה להזהיר להם מאד, או לחומר העניין או להיותם מורגלים לחטוא בהנה. I don't know that that is so, but perhaps I should consider all instances. I would just consider it an elaborate construction meaning "say to people X the following message." Ramban also cites the possibility that dabber in all those cases means gather them for a message, and so too here. I don't think this is necessary, as a matter of peshat.