One case that comes to mind is the Mishnah in BM 33a, where one must decide between "you are not able to hide" from taking your father's lost object and "you are not able to hide" from taking your Rebbi's lost object. One should take his Rebbi's unless his father is also a Talmid Chacham. It continues with other cases of deciding between positive commandments, but those aren't applicable for the sake of this question. Nevertheless, even this case is deciding between two applications of the same prohibition.
The Mishnah in Horayos 13a gives a few other cases, which are still applications of the same prohibition to two people: "you are not able to hide" of a man's lost object versus a woman's (man wins) and "do not stand [by idly] by your friend's blood [being spilled]" of a man versus a woman (man still wins), in addition to several positive commandments between man and woman (in which woman wins).
The tenth perek of Eiruvin opens up with several cases involving bringing lost pairs of tefillin inside the Eiruv on Shabbos; this is both "you are unable to hide" and disrespecting the Tefillin (which, although their place is unspecified in the Mishnah, may very well be on the ground or in a place where they will be ruined) versus violating Shabbos (of which the passuk says שמור, an expression of a negative commandment). The Mishnah paskens that Shabbos is more important, and one should either wear the Tefillin into the Eiruv and then take them off, or, if there are too many or if they're tangled (as untying is a violation of Shabbos) one must stay by them until after Shabbos and then carry them back inside.