The common minhag is for the whole congregation to stand up when the Aseres Hadibros are read in shul. However, the Rambam was very against this minhag and thought it should be abolished. The Rambam writes in a teshuva that the Aseres Hadibros used to be part of Davening, however the gemara in Berachos comments that Chazal abolished this so that people would not think that the Aseres Hadibros are more important then other parts of Torah. Therefore, the Rambam writes that standing up for the Aseres Hadibros should also not be done, as it promotes this idea that the Aseres Hadibros are more important then other pesukim.
RYBS explained the minhag as follows. He pointed out that the Rambam does not mention the Taam Haelyon (which is how we read the Aseres Hadibros). What is the difference between the Taam Haelyon and Taam Hatachton? The Taam Hatachton (the regular trup) splits up the dibros into pesukim. Some dibros are a few pesukim, and some pesukim contain a few dibros. The Taam Haelyon on the other hand, treats each dibra as 1 unit (as 1 pasuk). We can ask how is this allowed? After all, kol pesuka d'lo paska moshe anan lo paskinan (we are only allowed to read the Torah broken up in the original pesukim, we can not put 2 pesukim together or split a pasuk apart)? The Taam Haelyon wouild seem to violate this halacha.
RYBS answered that 1 question answers the other. He explained that when we read the Aseres Hadibros we are not reading it as Krias Hatorah, rather, we are trying to recreate Har Sinai. Therefore we read the Aseres Hadibros in the Taam Haelyon because that is how Hashem said it. Therefore, we also stand up, as we received the Torah at Har Sinai standing up. Since we are trying to recreate Har Sinai we are allowed to stand and split up the Pesukim differently, we are just recreating what happened. The Rambam on the other hand, since he did not have the Taam Haelyon, held that the reading of the Aseres Hadibros is just regular Krias Hatorah. Therefore it is inapropriate to stand up for them as they are no more important then any other pesukim and therefore he was against the minhag of standing up.