Question: May I arrange the dishes remaining on the table after a meal while reciting Birkat Hamazon?
Answer: The Gemara (Berachot 16a) states that workers constantly involved with their labor may not recite Birkat Hamazon while performing their labor. Maran Ha’Bet Yosef (end of Chapter 191) derives from this Gemara that one may not perform any work while reciting Birkat Hamazon. (The Talmud Yerushalmi writes this explicitly in Masechet Berachot, Chapter 5, Halacha 2.)
Several Rishonim as well as some Geonim in their responsa (see Toratan Shel Rishonim, Volume 1, page 56) write likewise that one may not be involved with any form of work while reciting Birkat Hamazon.
Indeed, the Turei Zahav writes, as follows: “Certainly, this applies to all other Mitzvot as well in that one should not be involved with discussing other matters (or doing other things) and they only mentioned this law regarding Birkat Hamazon for a different reason (mentioned in the Turei Zahav). However, one should not perform any Mitzvah while involved with one’s personal matters, for such behavior shows that the Mitzvah is being performed without concentration and intent and is being performed in a superficial manner which would be included in what the Torah writes, ‘And if you walk contrary unto me and will not wish to listen to me; I will bring seven times more plagues upon you according to your sins.’ This means that even if you will walk ‘with Hashem’, i.e. by performing His Mitzvot, nevertheless, if you perform them in a superficial and flippant manner, this is included in the above Torah prohibition.”
For this reason, the Poskim write that one may not even be involved in a light matter or action while reciting Birkat Hamazon. Thus, one may not dry one’s hands, put on one’s coat, or clear or arrange dishes during Birkat Hamazon (see Halacha Berura ibid).