What halakhic subtopics might be covered by The Tur and not by the Shulchan Aruch? I'm especially looking for examples where a teshuvah (or other halakhic opinion) relies on the Tur, at least in part, but not on the Sh"A (because the matter's not found in the Sh"A).
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The Rama at the very end of YD 331 (the chapter that discusses the laws of Terumot and Maaserot) notes that the Shulchan Aruch left out all the rules of Ma'aser 'Ani which are discussed in the Tur there. Presumably any further Halachik discussion about Ma'aser 'Ani is based on the Tur and not the Shulchan Aruch.
The Tur in YD 116 elaborates for several pages on the laws of liquids that were left revealed that are forbidden due to the concern that snakes drank from them and poisoned the water. Shulchan Aruch omits the entire discussion explaining that nowadays we do not live near such snakes. However, as the Pri Chodosh Points out, in places were snakes are common these laws do apply and one would have to refer to the Tur in such cases.