Gemara

(Mishnah - Beis Shamai): If someone ate and forgot to bless, he must return to where he ate and bless there;

Beis Hillel permit him to bless in his current place.

53b (Rav Zevid): They argue about one who forgot to bless. If he intentionally left without blessing, all agree that he must return to his place.

Objection: This is obvious. The Mishnah explicitly says that he forgot!

Answer: One might have thought that the same applies if he deliberately left without blessing. The Mishnah discusses when he forgot, in order to teach the extremity of Beis Shamai's opinion (even then, he must return);

Rav Zevid teaches that this is not so.

(Beraisa) Beis Hillel (to Beis Shamai): If one ate at the top of a building, must he go up again to bless?!

Beis Shamai: If he left his wallet there, he would return to get it for his own needs. All the more so he should return for the sake of Heaven!

One Talmid forgot to bless. He returned to his place, like Beis Shamai, and found a wallet full of gold coins;

Another Talmid deliberately left before blessing, and did [not return,] like Beis Hillel says [about one who forgot]. A lion ate him.

Rabah bar bar Chanah was traveling with a caravan. He forgot to bless. He reasoned that if he tells them, they will say "[we need not wait for you to return.] You can bless Hash-m anywhere!"

He told them that he forgot a golden dove. They agreed to wait for him. He returned, blessed, and found a golden dove.

52b: Beis Hillel forbids using an ignoramus waiter, and Beis Shamai permits.

(R. Yosi b'Rebbi Chanina): The Halachah follows Beis Hillel in our entire Perek, except for this law. Here, the Halachah follows Beis Shamai.

Rishonim

Rambam (Hilchos Berachos 4:1): One must bless Birkas ha'Mazon or me'Ein Shalosh in the place where he ate. If one forgot to bless, and remembered before the food was digested, he blesses in his current place. If he was Mezid (knowingly left without blessing), he returns to where he ate and blesses there. If he blessed in his current place, he was Yotzei.

Rosh (8:5): Rav Amram Gaon rules like Beis Shamai. Above, we said that in the entire Perek, the Halachah follows Beis Hillel! This rule does not teach about here, for even Beis Hillel agree that it is good to do like Beis Shamai, just they do not exert one to do so. This is only for Birkas ha'Mazon.

R. Yonah (40a DH Gemara): Here, the Talmid who was stringent like Beis Shamai was rewarded. Above (11a), R. Tarfon was Chayav Misah (b'Yedei Shamayim) for being stringent like Beis Shamai! There, even though Beis Hillel say 'we lean and read', they do not mean that one should intend to do so. Rather, if he was leaning, he may read leaning. If he was not leaning, and he intentionally leans, he transgresses Beis Hillel. Here, Beis Hillel agree that even if he forgot, it is praiseworthy to be stringent and return. The Talmid did not transgress Beis Hillel, therefore he was rewarded.

R. Yonah (DH ul'Inyan): Some Ge'onim and Rav Shmuel ben Chafni rule like Beis Shamai, that even one who forgot must return to his place. They learn from the Talmid who found a wallet because he did like Beis Shamai. This implies that one should do so. This is not a proof. We said that in the entire Perek, the Halachah follows Beis Hillel! The episode was not brought to reject Beis Hillel. Rather, it shows that one who toils to return to his place is rewarded for his toil.

Tosfos (Sukah 3a DH d'Amar): Rav Amram Gaon rules like Beis Shamai regarding one who left without blessing. Even though the Gemara said that the Halachah follows Beis Hillel in the entire Perek, perhaps we follow Beis Shamai because even Beis Hillel agrees that it is better to do like them.

Poskim

Shulchan Aruch (OC 184:1): One who ate in one place must bless before he leaves his place.

Source: Ran (Pesachim 20b DH Mesivei), cited in Beis Yosef (DH Kasav ha'Ran).

Rema: See above (178:2).

Shulchan Aruch (ibid.): If he left his place and did not bless, if he was Mezid, he returns to his place and blesses. If he blessed where he remembered, he was Yotzei.

Rema: This is only according to the Rambam. The Rosh holds that even if he was Shogeg, l'Chatchilah he returns to his place and blesses. If he was Mezid, even b'Di'eved he was not Yotzei if he did not return.

Beis Yosef (DH u'Mah): The Tur says that if he was Mezid and he blessed in his (new) place, he was not Yotzei. This is because he rules like Beis Shamai, that even if he was Shogeg he must return to his place. If even when he was Mezid, it is only l'Chatchilah to return to his place, what is the difference between Shogeg and Mezid? We must say that if he was Mezid and he blessed in his place, he was not Yotzei. The Rambam rules like Beis Hillel, that if he was Shogeg, he blesses where he remembered. It suffices to say that Mezid is one level more stringent, that l'Chatchilah he must return to his place. It is unreasonable to say that Mezid is two levels more stringent, that if he blessed in his place, even b'Di'eved he was not Yotzei.

Bach (DH v'Nir'eh): The Tur learned from the Rif and Rosh. Since the Halachah follows Beis Shamai, why did they bring Rav Zevid's teaching? It is relevant only to Beis Hillel, who distinguishes Shogeg from Mezid! Also, why did the Gemara say that one Talmid 'did b'Mezid like Beis Hillel', since Beis Hillel agree about Mezid? The Tur concluded that the Rif and Rosh brought Rav Zevid to teach that we learn from Beis Hillel to Beis Shamai. Just like Beis Hillel distinguishes Shogeg from Mezid l'Chatchilah, Beis Shamai distinguishes them b'Di'eved. One Talmid 'did b'Mezid like Beis Hillel', i.e. he left b'Mezid with intent to bless elsewhere, for he needed to go. After he blessed in the new place, he did not return to his initial place to bless there. This is like Beis Hillel, who hold that even if he was Mezid, he was Yotzei. This implies that according to Beis Shamai, he was not Yotzei even b'Di'eved. Mezid according to Beis Hillel is like Shogeg according to Beis Shamai.

Taz (1): What is the Tur's source to distinguish between Shogeg and Mezid regarding one who blessed where he remembered? Beis Shamai does not distinguish regarding the Chiyuv to return to his place. We can say that likewise they do not distinguish if one forgot and blessed where he remembered. The Beis Yosef said that surely there is a difference between Shogeg and Mezid. This is astounding. Surely, Beis Shamai does not distinguish regarding the Chiyuv to return. We can say the same regarding one who blessed where he remembered! It is unreasonable to say that in either case he was not Yotzei. If so, Beis Shamai should have taught this bigger Chidush.

Suggestion: Perhaps they argue about whether or not one must return to his place, to teach the extremity of Beis Hillel, that he need not return, because he was Shogeg.

Rejection (Taz): We hold like Beis Shamai. Had Rav Zevid not taught that Beis Hillel agree about Mezid, we would have said that (they argue even about Mezid, and) the Tana teaches the extremity of Beis Shamai. Surely Rav Zevid had a tradition that this is not the case. Therefore, elsewhere we surely say that the Tana teaches the extremity of Beis Shamai, since the Halachah follows them in this Halachah.

Suggestion: Perhaps he was Yotzei even b'Mezid if he blessed where he remembered.

Rejection (Taz): The episode with Rabah bar bar Chanah disproves this. If one is Yotzei b'Di'eved, why did he seek strategies to convince them to wait for him? This is the biggest b'Di'eved (a caravan that does not want to wait for him)! Rav Sheshes holds that one may pray while traveling, but once he was traveling with Rav Chisda, and Rav Chisda stood to pray, so Rav Sheshes did so also (30a). Even though he holds that it is better to pray standing, had his company not stopped to pray standing, he would have prayed while traveling. Rather, we must say that b'Mezid one is not Yotzei (if he blessed where he remembered).

Taz (1): The Rambam holds that even if he left b'Mezid and blessed where he remembered b'Mezid, he was Yotzei, for the Gemara says only l'Chatchilah that one must return. We should rule that Safek Berachos Lehakel, and b'Di'eved he was Yotzei in every case. If he did not bless yet, even if he was Shogeg he should return, if possible, unless his company will not allow this.

Gra (DH v'Im): We learn from the Talmid who acted like Beis Hillel. The words of the Gemara are difficult. Beis Hillel does not argue about Mezid! Rather, they argue also about Mezid b'Di'eved, just like they argue about Shogeg l'Chatchilah. Each of them are one level more stringent about Mezid than Shogeg.

Shulchan Aruch (ibid.): If he was Shogeg, the Rambam holds that he blesses in his current place, and R. Yonah and the Rosh say that also he returns to his place and blesses.

Kaf ha'Chayim (6): The Birkei Yosef (and Ma'amar Mordechai) say(s) that "R. Yonah" is a printing mistake, for R. Yonah himself rules like the Rambam.

Mishnah Berurah (6): This opinion agrees that it is praiseworthy to return, just he is not obligated.