Are two biological brothers allowed to marry two biological sisters? Are there any halachically issues involved with such a marriage? I do recall hearing something about it but not sure what it was.
In the Tzavaos of Rabbi Yehuda HaChasid #25 seen here he says two brothers should not marry two sisters.
See note #37 (#32 in the linked edition) from Rabbi Reuven Margolis quoting the Noda Biyehuda Even HaEzer 79 who brings cases in the gemara where we see this was not something they adhered to.
EDIT: To clarify the issue and for those who don't know, Seffer Chassidim was a work by one of the most famous Chasidei Ashkenaz. The ideas presented in the work are of a more mystical nature than standard halacha. However the ideas are most definitely referenced in later halachic works.
The section this quote is coming from is referred to as the Tzavaos, the instructions. This section presents an interesting dichotomy in being more often quoted (perhaps due to it's smaller size or seemingly imperative section name) but at the same time more easily absolved of the need to adhere to this section with the understanding that these instructions were meant for his descendants alone and no-one else.
This last idea was presented by the Noda BiYehuda and quoted often by subsequent authorities. But many times Rabbis only use this as an allowance for something they would have rather avoided. For instance, concerning some of the Tzavaos which people seem to care about such as not marrying a spouse with the same name as a parent, or making two marriages on the same day. Some of these ideas have been accepted as 'minhag' some have not and some have been allowed when there is a need. And of course as @Fred pointed out, there are many Tzavaos which are completely ignored.
Similarly, another famous idea from Rabbi Yehuda Hachassid is to not cut one's hair on Rosh Chodesh. This had been brought in the Magen Avraham and Mishna Berurah to keep. But there is a tradition in the Chafetz Chaim's family that he never meant to be stringent when Rosh Chodesh is on Friday which would cause one to enter Shabbos disheveled. Rabbi Yaakov Kaminetzky also had an inference from the Magen Avraham himself that he thought it was nice to take this idea into consideration, but ultimately in a situation of need, like Erev Shabbos, one does not need to worry about it.
This is the approach taken by Rabbi Margolios in that note in the name of various Rabbis starting with the Noda BiYehuda. He quotes the Gemara in Brachos 44a where we find 80 pairs of siblings married off to each other, seemingly a proof against this Seffer Chassidim, but he concludes that since we find their unfortunate demise in Yoma 38b, we cannot use this as proof to defend an allowance. He quotes as well from that gemara where we find the daughters of Rav Chisda marrying the brothers Rami bar Chama and Mar Ukva bar Chama, which again would seemingly imply the complete allowance of such unions. He deflects this though by pointing out that the gemara went out of its way to point out how great they were, something he claims is superfluous unless the gemara was attempting to give a good reason why these brothers were allowed to marry these sisters. He also adds on that even after finding a good reason why to allow this marriage, the end result was not a happy one as we find Rami bar Chama dying young in Bava Basra 12b where Rava marries his widow.
The note in my edition quotes more Tshuva Sefarim who discuss this, but does not go into details.
Practically speaking, this is seemingly not something people are stringent about. In another note there Rabbi Margolios quotes The Seffer Tzemach Tzedek on Yoreh Deah saying he married two of his children to siblings.
Yes they are allowed to marry. The problems are discussed here. I personally know of such couples (and also brother-sister, sister-brother cross-over) among the most ultra-orthodox chasidim.
The issue is discussed in Yevamot and yes it is legal and that is not disputed.
What is mentioned is where it leaves the state of a widow if one brother dies childless, and what happens if there is a younger 3rd brother.
If there are just 2 brothers then Yibum is impossible and because it is, then there is no need for Chalitzah either, the woman is free to marry who she chooses (even a Kohen as she did not go through Chalitzah).
(I cannot remember the halachah if there's a 3rd brother)
Subsequent Rabbis have not "forbidden" it but have discussed whether or not it is a good idea.
The gmara Brachos 44a mentions that 80 pairs of brothers married 80 pairs of sisters.
Shmuel and his brother Pinchas married sisters.
Sanhedrin 28b: Shmuel explains what types of brother in laws are disqualified from testifying "for example like me and Pinchas who are brothers and brother inlaws"...