At least Okinawan has nonstandard kana to precisely represent their sounds. For example, Okinawan has two types of う ([u] and [ʔu]), whose difference is important for the local people to distinguish their native words.
Ainu also has special katakana, although Ainu may not be Japonic.
There is always a way to approximate their words only using standard kana, anyway.
To the best of my knowledge, in dialects in Honshu/Shikoku/Kyushu, no special kana is used. Of course there are regional varieties only describable with IPA (e.g., う is nearer to [u] than to [ɯ] in Kansai dialect) or allophones found only in some dialects (e.g., yotsugana and bidakuon), but they are not distinguishing features. (Bidakuon have special characters, but they are more like phonetic symbols to me.)