I would translate that as "Nejez se svýma špinavýma rukama.".
I can feel some difference in the meaning of "with your hands dirty"="with your hands when they are dirty" and of "with your dirty hands"="your hands are dirty so do not eat with them". But some native English speakers can comment on that.
I think when 'with' means together (walk with me) or possessing (cat with black ears) it is translated 's'/'se' but when 'with' means the instrument used then you just put the instrument in the instrumental case. Here 'dirty hands' is the instrument used for eating so no 's'/'se'
Actually, there are two possibilities in Czech.
Nejez špinavýma rukama. Indeed the hands are like an instrument and you should not eat with them when they are dirty.
Nejez se špinavýma rukama. Don't eat while having your hands dirty.
I just checked the corpus and both are possible.