Gemara

(Mishnah): One may not burn Nikbarim (Isurei Hana'ah that should be buried), one may not bury Nisrafim (which should be burned);

R. Yehudah permits to be stringent and burn Nikbarim;

Chachamim: This is forbidden.

Pesachim 27a (Beraisa - R. Yehudah): A Kal va'Chomer obligates burning to fulfill Bi'ur Chametz;

Bal Yera'eh and Bal Yimatzei do not apply to Nosar, yet Nosar must be burned. They apply to Chametz, all the more so burning should be required!

Chachamim: You cannot make a Kal va'Chomer if the stringency leads to a leniency!

If someone has no wood (to burn Chametz), you exempt him from destroying it. "Tashbisu Se'or mi'Bateichem" obligates destroying it in any way you can!

Berachos 10b (Mishnah - R. Tarfon): I was on the road and lied down to say Shma, like Beis Shamai. I endangered myself due to bandits.

Chachamim: You were liable to die for transgressing Beis Hillel's opinion!

Rishonim

Rambam (Hilchos Pesulei ha'Mukdashim 19:14): One may not bury Nisrafim; and one may not burn Nikbarim. If one is stringent to burn them, he is lenient about the ashes, for the ashes of Nikbarim are forbidden.

Rambam (Hilchos Chametz 3:11): Bi'ur Chametz is through burning, or crumbling and throwing to the wind, or throwing to the sea. If one burned it after six hours on Erev Pesach, the coals are Asur b'Hana'ah, since the Chametz was Asur b'Hana'ah.

Pnei Yehoshua (Pesachim 27b DH Chametz): Our Gemara connotes that it is a stringency to burn Chametz. Even though Temurah 34a connotes that one may not burn Nikbarim, because their ashes are forbidden, R. Yehudah argues also there and permits to burn them. The Gemara connotes that he holds that the ashes are permitted. (If so, Rabanan did not need to refute his Kal va'Chomer, for according to them there is no Kal va'Chomer. In any case, they said that even according to his opinion, it is not a Kal va'Chomer.) Even though we hold that the ashes of Nikbarim are forbidden, for Chametz this is only mid'Rabanan. The Torah forbids Isurei Hana'ah only in the normal way to benefit from them. Surely, ashes of Chametz is not the normal Hana'ah! Further, I say that the Isur of the ashes of all Nikbarim is only mid'Rabanan, unlike Tosfos (33b DH ha'Nisrafim). This can resolve why the Rambam rules like Chachamim regarding Chametz, yet permits burning it.

Rosh (Berachos 3:2): If one lost a relative on Shabbos, and the burial will not be until tomorrow, and he did not say Havdalah on Motza'ei Shabbos, he may eat without Havdalah. We say (Berachos 17b) that one whose Mes is in front of him is exempt from all Mitzvos (Aseh) of the Torah. The Yerushalmi says that anyone exempt from a matter who does it, is called a Hedyot (commoner).

Hagahos Maimoniyos (Seder Tefilos, Nusach Berachos ha'Emtzayos, 4): We do not mention Rosh Chodesh in the Musafim of Rosh Hashanah. If one is exempt and fulfills, he is called a Hedyot.

Ramban (Kidushin 31a DH bi'Yerushalmi): The Yerushalmi says that anyone exempt from a matter who does it, is called a Hedyot. That is when he is not commanded at all. It is as if he adds to the Torah. If one does Mitzvos of the Torah properly, even though he was not commanded, e.g. women and Nochrim, they receive reward, for "its paths are pleasant, and all its ways are Shalom." We do not stop women from blowing a Shofar on Yom Tov, even though mid'Rabanan one may not blow without need, for this is considered necessary for a Mitzvah. Therefore, they may bless if they want. R. Tam holds like this.

Rosh (Shabbos 2:18): It is a good custom that women light the wick (for Ner Shabbos) and extinguish it, so it will light easily. However, the Halachah follows R. Akiva, who permits even if it is not singed.

Be'er Sheva (Teshuvah 21): Why does the Rosh praise the custom? R. Tarfon was liable for being stringent like Beis Shamai (Berachos 10b). The Tur and other Poskim say that one who was sitting may not be stringent to stand for Shma (in the morning). We rebuke him! Here also, the women do like R. Eliezer, who was from Beis Shamai! Some women are stringent not to eat meat for the three weeks between Tamuz 17 and Tish'ah b'Av, and add to refrain until the next Shabbos. The Maharshal (Teshuvah 54) criticizes them, for this is like R. Shimon ben Gamliel, and Rava rules unlike him (Ta'anis 30a)! The Maharshal brings a proof from R. Tarfon. Perhaps that case is different. Perush ha'Mishnayos explains that he endangered himself to descend from his animal and do like Beis Shamai, and unlike Beis Hillel. Beis Hillel permit to say Shma while on the animal. However, the Yerushalmi says that if one is exempt and fulfills, he is called a Hedyot. Why does the Rosh praise the custom? Tosfos said that women do so needlessly. Perhaps the Yerushalmi says that he is called a Hedyot only when he does so to be stringent. Here, the women do not intend to be stringent lest they come to tilt the Ner if it does not light well, like R. Eliezer. Rather, they intend that it burn well, for Shalom Bayis. Sefer ha'Terumah says so about Sirtut in Tefilin (scratching a line to ensure that one will write straight). He is not called a Hedyot if he does so in order to write straight, but not if he does so to be stringent.

Be'er Sheva (Teshuvah 21): I was always bothered by the teaching that if one is exempt and fulfills, he is called a Hedyot, and (Yerushalmi Nedarim 9:1) 'is it not enough what the Torah forbade' (that you add stringencies)! We have hundreds and thousands of stringencies! Chachamim commanded 'sanctify yourself with what is permitted to you', 'make a guarding for My guarding' (Yevamos 20a, 21a)' I wanted to say that one is called a Hedyot only in a matter that could result in an Isur. Then, there is no reason for a stringency or separation. However, the Yerushalmi asked why Rav Shmuel bar Yitzchak interrupted eating to pray, for one is exempt and fulfills is called a Hedyot, even though continuing to eat could result in Isur (omitting Tefilah)! Sefer ha'Terumah connotes that it applies only when an Isur could result, and he is stringent, but not regarding Sirtut and similar cases. Perhaps 'sanctify yourself with what is permitted to you' means only that one should not seek excess pleasures, but not to totally forbid what the Torah permitted, and similarly regarding 'make a guarding for My guarding.' This requires investigation.

Poskim

Shulchan Aruch (OC 445:1): Bi'ur Chametz is through burning, or crumbling and throwing to the wind, or throwing to the sea.

Magen Avraham (1): Our texts of the Gemara say that Chachamim permit even to crumble it. The texts of the Rif and Rosh connote they permit only crumbling it. The Tur questioned why the Rambam permits burning it. Perhaps the Gemara forbids burning it like Nisrafim, i.e. to benefit from the ashes. We do not decree lest he benefit from the ashes.

Gra (DH Sorfo): Even according to the texts that do not say 'even', Chachamim permit burning, for R. Yehudah says 'only burning.' Also, we cannot say that all the Mishnayos and Beraisos that mention burning Chametz are all like R. Yehudah, and R. Meir, R. Yosi, R. Shimon, R. Gamliel, R. Eliezer, R. Yehoshua and R. Eliezer Ish Bartusa all agree. (If so, we would not call this (only) R. Yehudah's opinion!) Rather, Chachamim agree that the primary Mitzvah is burning.

Rema: The custom is to burn it.

Taz (2): The custom to burn it is because there are different opinions about whom the Halachah follows. Therefore, we are stringent. Therefore, if one fulfills Bi'ur Chametz through burning after six hours, and afterwards found pieces of Chametz, he need not burn them. It suffices to destroy them, for after six hours we are lenient about Chametz like for other Isurim, e.g. any amount (does not forbid a mixture, unless it gives taste), or if it harms the taste (of a mixture, it does not forbid it). Even during Pesach we hold like Rabanan, just it is a custom to burn like R. Yehudah. Before the (first) night (of Pesach), if he already fulfilled the Mitzvah, he need not burn what he finds.

Gra (DH v'Tov): It is a stringency to burn it. R. Yehudah learns from Nosar, which is burned only during the day, and not on Yom Tov. He holds that also Chametz is burned only during the day.

Mishnah Berurah (6): The custom is to burn even during the proper time, i.e. the fifth hour on Erev Pesach, and all the more so after six hours or during Pesach. Then surely one should be stringent to burn it.

Kaf ha'Chayim (11): Also, Chametz alludes to Klipos (shells of Tum'ah). Burning alludes to removing Tum'ah and approaching the time of redemption.

Rema (639:7): If one was sleeping in a Sukah and any amount of rain came, he may leave. Anyone who is exempt from Sukah and does not leave does not receive reward. He is called a Hedyot.

Bi'ur Halachah (DH v'Chol): He is called a Hedyot only when there is an aspect of Isur, e.g. one who is in pain, for this desecrates Yom Tov. One must honor even Chol ha'Mo'ed. If he is not in pain, he may be stringent, like R. Gamliel.

Kaf ha'Chayim (92): The Acharonim ask that regarding many stringencies, the Poskim say that one who is stringent will be blessed. It seems that when there is an argument and the Halachah follows the lenient opinion, if the stringent opinion is zealousness and carefulness also according to the lenient opinion, then one who is stringent will be blessed. If not, he is called a Hedyot. The Ramban and Ritva say that he is called a Hedyot when he does a Mitzvah that everyone is exempt from. If some are obligated, e.g. men are commanded about Mitzvas Aseh sheha'Zeman Gerama and women are exempt, one who is exempt and fulfills is not called a Hedyot. This is why women may fulfill such Mitzvos. This refers to Mitzvos Bein Adam la'Makom. Regarding Bein Adam l'Chavero, we are commanded to go beyond the letter of the law, so we never say if one is exempt and fulfills, he is called a Hedyot.

Bach (OC 293:1): The Tur teaches that the custom is to delay Ma'ariv on Motza'ei Shabbos. This is not like if one is exempt and fulfills, who is called a Hedyot. Rather, he receives reward for adding from Chol to Kodesh. R. Yosi desired the portion in the world to come of those who delay doing Melachah on Motza'ei Shabbos, even though letter of the law it is permitted.

Bach (CM 294:11): The Tur teaches that if a deposit was stolen from a Shomer Chinam and he swore that he was not negligent, he may toil to find the thief. He is rewarded for this. We do not say that since he exempted himself through his oath, he has no more Mitzvah to return the Aveidah than anyone else. He is not like if one is exempt and fulfills, who is called a Hedyot, or one who gets involved in another's quarrel.