Is it assur for a Yisraeli or a Levi to marry a Bat Kohen? I know there are prohibitions for a Kohen when it comes to marriage but I was wondering if there are any special requirements for the husband of a Bat Kohen.
The Gemara in Pesachim 49A states that an Am Haeretz (Ignoramus) should not marry a Kohenes (daughter of Kohen). If they do, the marriage won't last, as one of them will die young or they will quarrel. This is also stated in Shulchan Aruch Even Haezer 2:8. It seems from Ramah in 2:6, that it may only refer to someone who is lax in keeping the mitzvos. Pischei Teshuvah quotes Chavas Yair, that today's religious Jews are not considered Ignoramus regarding this law.
No there isn't.
Gemara in Pesachim (49a) discusses two apparently contradictory sentences of Rabbi Yochanan, one which says that if an Israel who marry a Kohen's daughter, the couple does not succeed. In the second sentence, Rabbi Yochanan says that in way to be rich, it's good to marry the daughter of a Kohen. The conclusion is that the Israel who is Am haarets is not adapted to her, but a Talmid chacham is. This is not an Halacha of Yochasim, as the mitsvot of Kohanim a dressed below.
The Tur and Shulchan Aruch (in sayif 8) quoted the conclusion of the Gemara. We already soon that this rule is more an advise than a prohibition.
Strict laws of marriage:
Kohen women have not the same restrictions as men (e.g. regarding the prohibition to be unclean Bene Aharon and not Benot Aharon, see Rambam Isure Bia 19, 10 who uses this drasha, according to the Magid Mishne understanding, to allow chalal to marry with a divorced woman or a convert, so it is obviously true for a kohen woman not least than for a chalal to marry a convert). See in Sefer Hachinuch 266 - 268 who summarizes that special marriage rules concerning Kohanim are specific to males. In mitsva 269, it seems that the mitsva "to make him holy" is to honor kohen males only.
The Tur Even Haezer 2, in name of Rabenu Tam who writes explicitely that there is no prohibition for a kohen-woman to marry even a man who is chalal.
ורבינו תם פירש שאין בדיקה אלא לכהונה. ... אבל אשה כהנת אינה צריכה לבדוק אחר האיש, שלא הוזהרו כשרות לינשא לפסולין; אבל ליוחסין שאין בו צד כהונה, לא בעי בדיקה, עד כאן. Kohen women are not prohibited to marry pesulim likh'una.
The Bach is more explicit:
דאשה כהנת לא הוזהרה שלא ליקח חלל גר ומשוחרר אבל בקרא ערעור דפסול ממזרות פשיטא דאשה הוזהרה כמו האיש אדרבה פשטא דקרא לא יבא ממזר הוא שלא ישא ישראלית
A Kohen woman is not prohibited to take a Chalal, convert, or a freed slave, but the verses or prohibition to mamzerut are equivalent for men and women, the verse regarding prohibition of marrying with mamzerim is formulated as a prohibition of marriage between a man mamzer (bastard) and a kasher woman.