It could be expressed as the following:
Please do not (don´t) mix organic items (we are talking about organic waste, organic garbage, here.) with recyclables.
Expressed in that way would be bit informal, in a more business-like way, we could say "Organic items should not be mixed with recycables.", in a note from the waste collection and recycling services for example.
And in other ways, of course. It worth noticing that being organic material doesn´t necessarily means it isn´t non-recycable.