The custom of learning Torah the entire night of Shavuot is mentioned by the Magen Avraham (Orach Chaim #494), based on the Zohar: we dedicate the night to learning Torah in an attempt to rectify a mistake made by the Nation of Israel at the time of the Giving of the Torah. When Hashem “arrived” to give the Torah to the Nation of Israel, we were still sleeping and had to be woken up. The custom therefore developed to stay awake all night to spirituality make-up for our oversleeping and to show our zeal for the Torah. But one should be aware that if, on account of the exhaustion of learning Torah all night, he cannot daven Shacharit in the morning with proper concentration, it is better not to stay up since davening properly is a clear obligation (the Magen Avraham makes this exact point regarding staying up all night on Yom Kippur – see Orach Chaim 611:11).
In fact, Ha-Rav Yitzchak Zev Soloveitchik, the Brisker Rav, was surprised that people are so particular to stay awake the entire night of Shavuot, which is a custom, while on Pesach night, when there is a law to discuss the Exodus from Egypt until one is overcome by sleep, people are not so careful. And in the city of Brisk, people were not careful to follow the custom of staying awake the entire night of Shavuot, since why is this night different from all other night...? And also, learning on Shavuot night is not more important than learning during the day… (Uvdot Ve-Hanhagot Le-Beit Brisk vol. 2, p. 79).
And it is related in the book "Ha-Shakdan" (vol. 2, p. 240) that one of Ha-Rav Yosef Shalom Elyashiv's grandsons once asked him why he does not stay awake all night on Shavuot like everyone else, but follows his regular learning schedule of waking up at 2:00 AM to learn Torah. Rav Elyashiv explained that he calculated that if he changed his few hours of sleep on that night, he would not gain more time learning Torah - he would actually lose 15 minutes of learning! For a few precious minutes of learning Torah, he decided that it is preferable to go to sleep at the beginning of the night as usual.
Each person should therefore carefully consider if it is worthwhile for him to stay up all night since there is a concern that "his gain is offset by his loss."
1. Talit
One who wears Tzitzit all night should not recite a new blessing on it in the morning. One should try to hear the blessing said by someone who is obligated to recite it or have the Tzitzit in mind when he recites the blessing over his Talit (Shulchan Aruch, Orach Chaim 8:16 with Mishnah Berurah #42).
2. Netilat Yadayim
One should wash "Netilat Yadayim" without a blessing or hear it from someone who is obligated to recite it (Shulchan Aruch Ha-Rav 4:13). It is preferable to use the restroom as one is then obligated according to all opinions to wash "Netilat Yadayim." After washing "Netilat Yadayim," he should recite the blessing of "Al Netilat Yadayim" and "Asher Yatzar" (Shulchan Aruch, Orach Chaim 4:13 with Mishnah Berurah #27, 29, 30).
3. "Elohai Neshamah" and "Ha-Ma'avir Sheinah"
They should be recited without the ending of using Hashem's Name or be heard from someone who is obligated to recite them, since these blessings where established over the return of the soul and removal of sleep and neither of these occurred (Shulchan Aruch, Orach Chaim 47 with Mishnah Berurah #30 and Biur Halachah). If one sleeps a half an hour, one is obligated to recite these blessings (Shulchan Aruch, Orach Chaim 4:16 with Mishnah Berurah #34-35 and Biur Halachah).
4. "Ha-Noten Le-Yaef Koach"
One should recite this blessing even if he is very tired, since this blessing was not established for the person's individual state, but as a general praise of Hashem who created His world which includes the removal of tiredness (Shulchan Aruch, Orach Chaim 46 with Mishnah Berurah #22 and Shulchan Aruch, Orach Chaim 47 with Mishnah Berurah #28). Chasidim recite all of the morning blessings even if they remain awake all night (Shulchan Aruch Ha-Rav 47:7 and Siddur Chabad in the laws before the morning blessings and blessings over learning Torah).
5. Blessings over Learning Torah
There is a dispute whether these blessings should be recited if one remains awake all night. One option is that the morning before Shavuot, one make a condition that the blessings will be for the following day as well. One can also hear the blessings from someone who did sleep, with both individuals having in mind that the blessings will apply to both of them (Shulchan Aruch, Orach Chaim 47 with Mishnah Berurah #25-28). If neither of these is an option, one can recite the blessings based on the opinion of the Shut Sha'agat Aryeh (#24-25) that these blessings are a Torah Mitzvah and in the case of a doubt, one is strict to recite them. This ruling is found in Maran Ha-Rav Kook's commentary on the siddur "Olat Re'eiyah" (vol. 1, p. 59 #5) and in Ha-Rav Ovadiah Yosef's responsa (Shut Yabia Omer vol. 5, Orach Chaim #6 and Shut Yechaveh Daat 3:33).
In this regard, women are also required to recite the blessings over learning Torah and these blessings are printed in all of the Siddurim for women. But how can they recite the blessing "Blessed is Hashem…who has made us holy and commanded us to engage in words of Torah" when they are not obligated to learn Torah? There are various answers, but the answer of Ha-Rav Yitzchak Zev Soloveitchik, known as the Griz, on the Rambam (at the end of Hilchot Berachot, p. 10) and Maran Ha-Rav Kook (Orach Mishpat 11, 2) is that these are not blessings over performing a mitzvah but blessings of praise. If the Torah was not given, the world would be in darkness for both men and women. Women therefore also thank Hashem for the Torah being in the world
