If a woman has a boy, and goes to the mikvah 14 days after she has the baby, even though she is still bleeding. Assuming the couple won't have relations, are they "over" any aveira?
The woman in this case has the status of a zavah on a rabbinic level. According to Rabbeinu Tam, affectionate contact with such a woman (even without engaging in marital relations) could be the cause of suffering an early death.
For all the details, see below the line.
On a biblical level, a woman who has given birth (yoledet) can go to the mikvah seven days after giving birth to a boy or fourteen days after giving birth to a girl (as long as she was not a zavah at the time of the birth - see below for details). She is then permitted to her husband until the end of thirty-three days following the birth of a boy or sixty-six days following the birth of a girl, even if she continues to bleed:
כָּל דָּם שֶׁתִּרְאֶה הַיּוֹלֶדֶת בְּתוֹךְ ל''ג שֶׁל זָכָר וְס''ו שֶׁל נְקֵבָה הוּא הַנִּקְרָא דַּם טֹהַר. וְאֵין מוֹנֵעַ אֶת הָאִשָּׁה מִבַּעְלָהּ אֶלָּא טוֹבֶלֶת אַחַר שִׁבְעָה לְזָכָר וְאַחַר אַרְבָּעָה עָשָׂר לִנִקֵבָה וּמִשַׁמֵּשֵׁת מִטָּתָהּ אַף עַל פִּי שֶׁהַדָּם שׁוֹתֵת וְיוֹרֵד:
Any blood that the yoledet sees within the thirty-three days for a boy or the sixty-six days for a girl is called 'blood of purity' (dam tohar). It does not prevent a woman from being with her husband; rather, she immerses seven days after having a boy or fourteen days after having a girl, and can then engage in marital relations, even if she continues to bleed heavily.
However, if she was a zavah at the time of the birth, she is classified as a yoledet bezov, and the rules are different:
אִם רָאֲתָה דָּם שְׁלֹשָׁה יָמִים אוֹ יֶתֶר בִּימֵי זִיבָתָהּ בְּצַעַר וַחֲבָלִים וּפָסַק לָהּ הַצַּעַר וְרָוַח לָהּ מִן הַחֲבָלִים אַחַר הַשְּׁלֹשָׁה יָמִים וְעָמְדָה בְּנַחַת כ''ד שָׁעוֹת אוֹ יֶתֶר אַף עַל פִּי שֶׁלֹּא פָּסַק הַדָּם וְאַף עַל פִּי שֶׁחָזַר הַצַּעַר וְהַחֲבָלִים אַחַר כ''ד שָׁעוֹת הֲרֵי זוֹ זָבָה. ... וְאִם יָלְדָה אַחֲרֵי כֵן הֲרֵי זוֹ יוֹלֶדֶת בְּזוֹב:
If she saw blood for three or more days during her zivah-days [when she is not expected to have her period] with pain and contractions, and then the pain and contractions stopped after the three days for twenty-four hours or more (even if the bleeding did not stop and even if the pain and contractions returned after twenty-four hours) she is a zavah ... and if she gave birth after that, she is a yoledet bezov.
כֵּיצַד דִּין יוֹלֶדֶת בְּזוֹב. צְרִיכָה לֵישֵׁב שִׁבְעָה יָמִים נְקִיִּים וְטוֹבֶלֶת לָעֶרֶב. וְאַחַר כָּךְ תִּהְיֶה מֻתֶּרֶת לְבַעְלָהּ וְאַחַר כָּךְ יִהְיֶה לָהּ דַּם טֹהַר
What is the rule of a yoledet bezov? She needs to observe seven clean days and then she immerses in the evening. After that she is permitted to her husband, and then she is subject to dam tohar.
יוֹלֶדֶת בְּזוֹב שֶׁלֹּא פָּסַק דָּמָהּ אֵין לָהּ דַּם טֹהַר. אֶלָּא כָּל דָּם שֶׁתִּרְאֶה כְּדַם זִיבָה הוּא
A yoledet bezov who does not stop bleeding does not have dam tohar. Rather, any blood that she sees is considered to be zivah-blood.
One final point of introduction, and then we can move on to your actual question.
Rabbinically, we treat every yoledet nowadays as a yoledet bezov:
כָּל הַיּוֹלֶדֶת בַּזְּמַן הַזֶּה הֲרֵי הִיא כְּיוֹלֶדֶת בְּזוֹב וּצְרִיכָה שִׁבְעַת יָמִים נְקִיִּים
Every yoledet nowadays is [considered] a yoledet bezov, and needs seven clean days.
A yoledet who goes to the mikvah nowadays while still bleeding
Now, on to your question. The woman in this case is unlikely to be a genuine yoledet bezov (on a biblical level) so bibically she would be permitted to her husband after her immersion. (In fact, this would be true here if she had immersed even seven days after the birth, since in the case under discussion she gave birth to a boy.)
Rabbinically, however, she is considered to be a yoledet bezov. I'm assuming (although you don't state so explicitly in your question) that she has been bleeding throughout the fourteen days prior to immersing, so rabbincally the immersion would not be valid, as she has not observed seven clean days.
Thus, this question is essentially a subset of this one. The reason I say a subset is because you note that the couple will not be engaging in marital relations. I assume that you are implying that they will not be observing any of the harchakot.
So we can re-state your question as: "Is there any transgression involved in not observing harchakot with a rabbinic zavah?"
תני דבי אליהו מעשה בתלמיד אחד ששנה הרבה וקרא הרבה ושימש תלמידי חכמים הרבה ומת בחצי ימיו והיתה אשתו נוטלת תפיליו ומחזרתם בבתי כנסיות ובבתי מדרשות ואמרה להם כתיב בתורה כי הוא חייך ואורך ימיך בעלי ששנה הרבה וקרא הרבה ושימש תלמידי חכמים הרבה מפני מה מת בחצי ימיו ולא היה אדם מחזירה דבר פעם אחת נתארחתי אצלה והיתה מסיחה כל אותו מאורע ואמרתי לה בתי בימי נדותך מה הוא אצלך אמרה לי חס ושלום אפילו באצבע קטנה לא נגע [בי] בימי לבוניך מהו אצלך אכל עמי ושתה עמי וישן עמי בקירוב בשר ולא עלתה דעתו על דבר אחר ואמרתי לה ברוך המקום שהרגו שלא נשא פנים לתורה שהרי אמרה תורה ואל אשה בנדת טומאתה לא תקרב כי אתא רב דימי אמר מטה חדא הואי במערבא אמרי אמר רב יצחק בר יוסף סינר מפסיק בינו לבינה:
The Sage in the school of Eliyahu taught a baraita that deals with this halakha: There was an incident involving one student who studied much Mishna and read much Bible, and served Torah scholars extensively, studying Torah from them, and, nevertheless, died at half his days, half his life expectancy. His wife in her bitterness would take his phylacteries and go around with them to synagogues and study halls, and she said to the Sages: It is written in the Torah: “For it is your life and the length of your days” (Deuteronomy 30:20). If so, my husband who studied much Mishna, and read much Bible, and served Torah scholars extensively, why did he die at half his days? Where is the length of days promised him in the verse? No one would respond to her astonishment at all. Eliyahu said: One time I was a guest in her house, and she was relating that entire event with regard to the death of her husband. And I said to her: My daughter, during the period of your menstruation, how did he act toward you? She said to me: Heaven forbid, he did not touch me even with his little finger. And I asked her: In the days of your white garments, after the menstrual flow ended, and you were just counting clean days, how did he act toward you then? She said to me: He ate with me, and drank with me, and slept with me with bodily contact and, however, it did not enter his mind about something else, i.e., conjugal relations. And I said to her: Blessed is the Omnipresent who killed him for this sin, as your husband did not show respect to the Torah. The Torah said: “And to a woman in the separation of her impurity you should not approach” (Leviticus 18:19), even mere affectionate contact is prohibited. The Gemara relates that when Rav Dimi came from Eretz Yisrael to Babylonia, he said: That student did not actually sleep with her with bodily contact; rather, it was in one bed that they slept without contact. In the West, in Eretz Yisrael, they say that Rav Yitzḥak bar Yosef said: When they would sleep together in one bed, she wore a belt [sinar] from the waist down that would separate between him and her. Nevertheless, since the matter is prohibited, that student was punished.
(Translation and elucidation from sefaria.org)
Rabbeinu Tam notes in Tosafot there (s.v. bimei livuneich):
שהיו רגילים לטבול שתי טבילות אחת לסוף שבעה לראייתה שהיא טהורה מדאורייתא בהך טבילה ואחת לסוף ימי ליבון לכך היה מיקל אותו האיש
They used to immerse twice, once seven days after the start of the period so that she would be pure biblically from that immersion, and once at the end of her clean days, which is why that man was lenient.
So we see that, at least according to Rabbeinu Tam, non-observance of the harchakot in the case described in this question could be the cause of early death (rachmana litzlan).
While these days everyone waits until they can get a clean hefsek and Shiva Nikiyyim, I'm not 100% sure that's an absolute Halachik requirement. M'Diroysa she is only tamei for 7 days. The next 33 days any blood is tahor. I believe Chumras R' Zeira only applies to blood that is inherently tamei.
The rabanun said every woman that gives birth is Yoledes B'Zov. Even though a Zava Gedola needs Shivaa Nikiyyim, the woman in our case can wait 7 days, and the blood she sees shouldn't invalidate the Shivaa Nikiyyim, as it is inherently Tahor.
The Geonim made a cherem on anyone who has relations after seeing dam tahor, here the couple isn't having relations.
P.S. While there is a minhag not to consider any blood dam tahor these days, I believe this is a case of "Yatzah Scharo B'Hefsaido". This minhag goes against the Hashkafa that Hashem doesn't want couples to have a lengthy period of separation. Unfortunately many couples these days end up touching, which is far worse. I'm not a Rav but I do plan on asking a shayla regarding this.
here are a few sources that discuss Dam Tahor: