As already indicated, "sympathetic buzz" is normal and in most cases it is not noticed in the final mix.
Detuning the snare drum or detuning the mounted toms (to get some seperation of tone) is the normal way to reduce buzz. Other options can be to place something between the head and the snare (not my preferred choice) or hang a think piece of fabric (like a bath towel) between the mounted tom and the snare.
Tuning should be your first choice. However, trust your ears, if you don't hear the snare buzz in your recorded mix, than don't worry too much about it - it's part of the instrument.