Gemara

(Mishnah): We do not whiten with an (iron) builder's trowel (rather, with a wooden one), lest it touch the stones and disqualify them.

Iron was created (with potential) to shorten life (through killing). The Mizbe'ach extends life. It is not proper that what shortens (life) touch what extends.

Berachos 8a: R. Yochanan was amazed that there are old people in Bavel, for it says, "Lema'an Yirbu Yemeichem vi'Yemei Veneichem Al ha'Adamah..." (long life is only in Eretz Yisrael).

When he heard that they come early to Beis ha'Keneses and stay late, he understood that this is their merit for long life.

55a (Rav Yehudah): If one prolongs (his meal at) his table, his life will be prolonged;

This gives more opportunity for poor people to come and receive food;

Question: "Ha'Mizbe'ach Etz... Zeh ha'Shulchan Asher Lifnei Hash-m" -- first it calls it Mizbe'ach, and then it calls it Shulchan (table)!

Answer (R. Yochanan and R. Elazar): When the Mikdash stands, the Mizbe'ach atones for the sins of Bnei Yisrael. After the Churban, (hospitality done on) one's table atones.

Rishonim

Shibolei ha'Leket (155): The custom is to remove the knife from the table at the time of Berachah (after eating). Some spread a cloth over it. They explain that prolonging one's (meal at his) table prolongs life, and after the Churban, a man's table atones for him. It says about the Mizbe'ach "Lo Sanif Aleihem Barzel", because the Mizbe'ach atones for man and extends life. Here also, the table extends life and the knife shortens. What shortens should not be on what extends. I heard another reason from R. Simchah. Once, a person was blessing Birkas ha'Mazon, and when he reached Boneh Yerushalayim he remembered the Churban, and he took the knife and stabbed himself through the heart. Therefore, the custom is to remove the knife at the time of Birkas ha'Mazon.

Poskim

Shulchan Aruch (OC 180:5): The custom is to cover the knife at the time of Birkas ha'Mazon. The custom is not to cover it on Shabbos and Yom Tov.

Beis Yosef (DH Kasav): The Roke'ach (332) and Avudraham (p.320) bring that we cover it because nowadays the table atones like the Mizbe'ach. R. Simchah says that it is because once, someone stabbed himself. Orchos Chayim brought both reasons. The custom is not to cover it on Shabbos and Yom Tov. According to R. Simchah, there is no reason to distinguish. In any case, the custom of Yisrael is Torah.

Taz (1): Why did the distinction bother the Beis Yosef only according to R. Simchah, but not according to the other reason? Perhaps he means that even according to R. Simchah it is difficult, and all the more so according to the first reason. R. Simchah could say that people are not so pained on Shabbos and Yom Tov. I say that it is not difficult for R. Simchah. The decree was like the episode. We find that Chazal warned not to wear a wooden shoe on Shabbos due to an episode (Shabbos 60a). They decreed only like the episode, which was with a wooden shoe (on Shabbos). Chachamim decreed not to carry Mei Chatas on a boat, due to an episode on a boat (Yevamos 116b). Also here, the episode was on a weekday, so they decreed only on weekdays.

Magen Avraham (4): According to the first reason, since we do not build the Mizbe'ach on Shabbos and there is no allusion to the Mizbe'ach (one need not cover it on Shabbos). Also, this applies only to an iron knife. The Levush does not distinguish between weekdays and Shabbos. Shulchan Shel Arba says that one should cover the bread at the time of Birkas ha'Mazon, lest it see its shame (that we bless over a cup of wine). See Siman 473:7.

Kaf ha'Chayim (16): There, the Rema says that when we lift the cup (at the Seder) we cover the bread. The Magen Avraham explains that it is to spare it from shame. If so, likewise one must cover the bread while saying Birkas ha'Mazon over a cup. However, one can distinguish. There, we say praise over wine, but not over bread. Here, the cup is only because we ate and blessed Birkas ha'Mazon on the bread, so there is no shame. Rather, the bread should be exposed to show his Simchah! Also Bigdei Yesha rejected Shulchan Shel Arba. Ma'amar Mordechai rejected him for another reason, and said that the custom is unlike him. Birkei Yosef (2) said that there is no concern for shame to the bread, since we blessed ha'Motzi on it. In Shirei Berachah (1), he brought from Pesikta Zutresi that they used to cover the table (at the time of Birkas ha'Mazon) to honor the Berachah, but the bread would be on top, so Berachah will rest on it. Beer Esek (27) gave reasons why the bread should be covered, but it seems that it should be exposed, and this is the custom.

Eliyahu Rabah (6): Mateh Moshe (304) says that all of the letters of the Aleph-Beis are in Birkas ha'Mazon, except for final Fei. This is because the angel Af Shatzaf Katzaf does not rule over a meal at which they bless Birkas ha'Mazon in the proper time, with intent. We cover the knife to allude to this, that at the time of Birkas ha'Mazon, the knife of the angel of death does not rule. Sefer Chasidim (102) forbids killing lice on a table on which people eat. If one does, it is like killing on the Mizbe'ach.

Eliyahu Rabah (7): Tola'as Yakov (Sod Birkas ha'Mazon) says that one may not use iron on the Mizbe'ach, because iron is the power of Esav. There is no Satan on Shabbos, and no bad occurrences, and those forces cease, therefore we do not remove the knife. He also says that on Shabbos we do not recall the Churban of the Beis ha'Mikdash, only the rebuilding. (Perhaps he refers to the Halachah that one leaves a place on the table empty, to recall the Churban. We do so on weekdays, but not on Shabbos (Mishnah Berurah 560:5) - PF.) Therefore, there is no concern (lest one stab himself on Shabbos). This answers the Beis Yosef's question.

Birkei Yosef (4): The Ramak says that one should not cover the knife on Shabbos or Yom Tov. Also Mahara Azulai says so, for Shabbos and Yom Tov allude to the world to come. Then, the Kelipos (shells of Tum'ah) and death will cease forever. The Eliyahu Rabah properly answered the Beis Yosef's question.

Kaf ha'Chayim (15): Mateh Moshe gave another reason. Yakov was blessed "mi'Tal ha'Shamayim umi'Shmanei ha'Aretz", and Esav was blessed "Al Charbecha Tichyeh." In Birkas ha'Mazon we mention Birkas ha'Aretz, which is Yakov's Berachah. Therefore we cover the knife, lest the staff of Resha'im rest in the portion of Tzadikim. I add that in Birkas ha'Mazon we pray for the rebuilding of Yerushalayim. Then, "Lo Yisa Goy El Goy Cherev." We cover the knife to hint that with Hash-m's help our request will be granted for the rebuilding of Yerushalayim, and swords will not be seen anymore. The Ari Zal says that if someone (i.e. his Neshamah; biologically, we all descend from Shes - PF) is from the source of Kayin, it is not enough to cover the knife. He must remove it from the table. We do not know who is from Kayin and who is from Hevel, therefore we all must remove the knife. If so, one must do so also on Shabbos.

Shulchan Aruch (151:6): Some forbid entering a Beis ha'Keneses with a long knife.

Source (Beis Yosef DH u'Mutar): Orchos Chayim brings so from Maharam, because Tefilah extends life, and the knife shortens life. One may not enter even if the knife is in his pocket, just like one may not enter Har ha'Bayis with his moneybelt. R. Peretz says that we are concerned only for Giluy ha'Rosh (an exposed head).

Taz (2): It seems that we are not concerned for a short knife. In Siman 180, we forbid any knife! The Beis Yosef brought both of them from Orchos Chayim. It seems that Birkas ha'Mazon is different, for it suffices to cover the knife. One can do so for any knife. Regarding a Beis ha'Keneses, he will need to leave it (elsewhere). This is a big exertion to have to go back to get it. Also, sometimes he needs it. Therefore, Chachamim were stringent only for a big knife, for which there is not so much need.

Mishbetzos Zahav and Kaf ha'Chayim (42): A short knife can be covered. A long knife is on his belt, and even if he covers it, it is evident.

Mishbetzos Zahav: Kolbo says that R. Peretz is stringent only 'b'Galuy', i.e. if the knife is exposed. According to the Beis Yosef's text of R. Peretz, even a covered knife is forbidden.

Kaf ha'Chayim (41): The Shulchan Aruch says 'some say' because an individual said it, but no one argues with it. (I.e. R. Peretz is not concerned only when it is in his pocket - PF.)

Gra (DH b'Sachin): Also the Beis ha'Keneses lengthens man's life, like it says in Berachos.

Mishnah Berurah (22): Eliyahu Rabah is stringent only if the knife is exposed.

Bi'ur Halachah (DH b'Sachin): Presumably, we are stringent for Stam people, but not for Chachamim who may eat and drink there, and they need a knife. This is like at a meal; we must cover the knife only at the time of Birkas ha'Mazon.

Kaf ha'Chayim (91:29): Based on Kabalah, one should not pray with anything of iron on him.

Ginzei ha'Kodesh (p.271:1, citing ha'Gaon R. C. Kanievsky Shlita): One should not put anything of iron, even a pen or ruler, on top of a Sefer.