The Gemora says that an Asham Talui must be brought if the Yavam and Yavamah did not wait the required 3 months after her husband was Niftar
This is because it could have been a 9 month baby from the 1st husband or a 7 month baby from the Yavam.
However why should a Korban be brought at all since we have a rule that "Ta'ah B'Dvar Mitzvah" is Patur from a Korban
And 1 opinion was that even if the mistake did not result in a Mitzvah, he is still Patur.
And a Kesiva V'Chasima Tovah
1) The Maharsha, at the beginning of Pesachim 72a, writes that "Ta'ah b'Dvar Mitzvah" applies only when, for example, someone thought that this was a Korban Pesach but in fact it was a different Korban, or a case similar to that of the Mishnah in Shabbos 137a, where there were two babies who needed a Bris, one after Shabbos and one on Shabbos, and the Mohel mixed up and circumcised on Shabbos the one that should have been circumcised on Sunday. The Maharsha writes that if the person thought that one is permitted to slaughter other Korbanos with the intention of it being a Korban Pesach, this would not be considered Ta'ah b'Dvar Mitzvah, but would be considered Shogeg and requires the atonement of a Korban.
We learn from the Maharsha that if one makes a mistake about the Halachah, he is not considered Ta'ah b'Dvar Mitzvah. In our case, the Yavam and Yevamah presumably did not know that the Halachah states that one must wait three months, so this is not considered Ta'ah b'Dvar Mitzvah.
2) I found an opinion in the Acharonim that Ta'ah b'Dvar Mitzvah applies even if the mistake was due to an error concerning the Halachah. This is the opinion of the Kapos Temarim to Sukah 43a and the Yom Teru'ah (by the same author) to Rosh Hashanah 29b.
a) The discussion begins with the Mishnah in Sukah (41b) where Rebbi Yosi says that if the first day of Sukos falls on Shabbos and one forgot and carried the Lulav in the public domain, he is exempt. Rashi (DH she'Hotzi'o) explains that Rebbi Yosi maintains that if one is Ta'ah b'Dvar Mitzvah and was Shogeg in a matter of Kares, he is exempt from a Korban.
b) The Kapos Temarim in Sukah (beginning of 43a) cites the question that Rashi and Tosfos ask there (DH v'Ya'avirenu): Why does the Gemara say that the reason one is not allowed to shake the Lulav on Shabbos is that he might come to carry the Lulav four Amos in the public domain? The Gemara should mention a different problem -- that one might come to carry the Lulav out from his private domain to the public domain!
c) The Kapos Temarim gives a different answer to this question than the answers given by Rashi and Tosfos. He makes a distinction between the prohibiton of carrying from one domain to another and the prohibition of carrying four Amos in Reshus ha'Rabim. The prohibition of taking from one domain to another is derived from verses in Shabbos 96b. In contrast, the Gemara there states that the prohibition against carrying four Amos in Reshus ha'Rabim is "Gemara Gemiri Lah" -- it is a Halachah l'Moshe mi'Sinai. The Kapos Temarim writes that everyone knows a Halachah derived from verses, but not everyone knows a Halachah l'Moshe mi'Sinai, which is not even hinted to in the written Torah. Therefore, Chazal were concerned that people might carry four Amos in Reshus ha'Rabim, but they were not afraid that they would carry from one domain to another. One sees from the Kapos Temarim that he learns that Ta'ah b'Dvar Mitzvah also includes mistakes in the Halachah.
(See Teshuvos Maharsham 6:39, in the emendation printed at the bottom of the page.)
d) According to this, it seems that we will have to say that the Yavam and Yevamah neglected to wait three months not because they did not know that there is such a Halachah, but rather they knew the Halachah but still did Yibum, which is why they must bring an Asham Taluy.
3) Here are more reasons for why doing Yibum before waiting three months after the death of the brother cannot be considered as Ta'ah b'Dvar Mitzvah:
a) The Gemara in Pesachim (72b) cites the Mishnah in Sukah (41b, mentioned in our previous reply) in which Rebbi Yosi says that if the first day of Sukos falls on Shabbos and one forgets that it is Shabbos and he carries the Lulav in Reshus ha'Rabim in order to perform the Mitzvah, he is exempt from a Korban because this was a mistake involving the performance of the Mitzvah of Lulav. The Gemara suggests that the reason why one is exempt in this case is that one is pressed for time. Rashi (DH d'Zmano) writes that he has to shake the Lulav today, so he quickly goes to an expert who can teach him the Halachos of Lulav.
The Gemara adds that according to Rebbi Yehoshua in the Mishnah in Pesachim (70b), who maintains that if one slaughters, on Shabbos, an animal that was consecrated to be a Korban other than the Korban Pesach, with intent that it is his Korban Pesach, and that animal was fit to be a Korban Pesach (i.e., a one-year-old sheep), he is exempt. The Gemara suggests that the reason why he is exempt according to Rebbi Yosi is that he is in a hurry to slaughter the Korban Pesach. Rashi (72b, DH Zmano) writes that he must slaughter the Korban Pesach today, before the time expires.
We see from the above cases that Ta'ah b'Dvar Mitzvah applies only if one is in a hurry to perform a Mitzvah and working against a time schedule, and this time pressure is why the person made a mistake. This would not apply to the Mitzvah of Yibum.
b) The Or Same'ach (Hilchos Shabbos 1:8, at the end) writes that Ta'ah b'Dvar Mitzvah does not apply to Mitzvos Bein Adam la'Chaveiro. The proof for this is the Gemara in Shabbos 30a, where Rebbi Yehudah says that if one extinguishes a candle on Shabbos for a sick person whose life is not in danger, one is liable for a Chatas. Why do we not say that he is making a mistake in performing the Mitzvah of visiting the sick? We learn from there that Ta'ah b'Dvar Mitzvah does not apply to Mitzvos one does for other people.
The Or Same'ach writes that this also explains why the Tosfos Yom Tov writes, on the first Mishnah in Maseches Shabbos, that Ta'ah b'Dvar Mitzvah does not apply when the homeowner gives a loaf of bread to the pauper standing outside. Why do we not say that this is Ta'ah b'Dvar Mitzvah since he is performing the Mitzvah of charity, even though he does a Torah prohibition of carrying from his house out into the street? The Tosfos Yom Tov writes that for the Mitzvah of Tzedakah there is no exemption. According to the Or Same'ach, who says that Ta'ah b'Dvar Mitzvah does not apply for Mitzvos done for other people, we now understand the reasoning of the Tosfos Yom Tov.
According to this, it would seem that Ta'ah b'Dvar Mitzvah does not apply to the Mitzvah of Yibum, since the purpose of Yibum is to perpetuate the name of the dead brother (see Devarim 25:7). This is a Mitzvah he performs for his brother, so it is a Mitzvah Bein Adam la'Chaveiro.
Kesivah v'Chasimah Tovah,