What is the source for the prohibition against Nivul Peh (using foul language)? Is it Biblical? Rabbinic? "Asmachta" (based on a principle derived from the Torah but ultimately not considered a Biblical mandate, ie., not a Mitzvah)? And based on the answer to the above, how severe is the prohibition (when can it be overridden, what, if anything, is the penalty, etc.)?
Therefore the Lord will take no pleasure in the young men, nor will he pity the fatherless and widows, for everyone is ungodly and wicked, every mouth speaks vileness. And for all this, his anger is not turned away, his hand is still upraised.
As @GershonGold pointed out, the Talmud (Shabbos 33a) derives from here that nivul peh is a very bad thing. It would seem strange that a Prophet would be decrying a certain behavior, especially in such terms, if it only went against a decree of the rabbinic courts, and not something defined as wrong by God himself. If we can't find an allusion to the wrongfulness of nivul peh in the Torah (i.e. the Pentateuch) then we should look harder. From textual context alone I don't think it makes sense to equate this with lighting Chanuka candles or muktza.
Yerushalmi Terumoth (ch. 1):
ולא יראה בך ערות דבר (דברים כג): ערות דיבור זה - ניבול - פה
Also, see Mesilat Yesharim (ch. 11) about the severity of this sin, where he brings this and other sources.
שבת לג ע"א says that due to the sin of Nivul Peh many Tzaros and bad Gezeiros happen and young people pass away. So I guess it is Rabbinic in nature.
I cannot find mention of nibbul peh in Rambam's Mishneh Torah, or in other lists of the mitsvot, such as Yereim, Semag, Semak, Ramban's addenda to Sefer Hamitsvot, etc. I therefore assume that according to Rambam, et al. it is not a Biblical prohibition and that the associated derashot are asmakhtot. I similarly find no mention if it in the major codes such as the Arba'ah Turim and the Shulhan Arukh. Their combined silence seems to imply that it is not a formal rabbinic prohibition either. That does not mean, however, that performing it is permissible, or even that doing so does not violate a rabbinic, or even biblical commandment. It just means that it is not its own legal category (in spite of its distinct name).
For example, cannibalism and public nudity can both be extremely problematic, even-though it is likely that neither is the subject of a discrete prohibition.