What do traditional Jewish sources say about Midrash Tanchuma (in contradistinction to the Midrash Tanchuma Yashan) regarding its date of redaction and authority vis-à-vis works of Chazal (such as other Midrashim).
It says in Shem HaGadolim of the Chida (Section 1 on Rabbi Yehudah Ebas and in Section 2 on Sefarim quoting Rabbi Di Lonzano) that the Midrash Tancuma we have today is not the original Tanchuma mentioned by Rishonim and before. It is a later redaction that according to some is what is left of Lamdenu.
According to many today, 'Midrash Lamdenu' has also been lost. So it is possible that the 'Lamdenu' mentioned by the Chida is some other, later text.
Midrashim in general are nice and a good souce for Divrei Tora for the Shabes table. However one cannot bring the words of the Midrash as Halacha, unless Chazal or one of the Rishonim specifically referred to that midrash as a source of that specific Halacha.
Rashi (and many others) commonly bring Midrash Tanchuma to explain verses in the Tora, as well as a Halachic authority.
Rav Tanhuma bar Abba himslef is an Amoira so he is in fact a member of Chazal. There are however some opinions as to who assembled the Ma'amarim into the one Midrash and the date at which it was conducted. This fact greatly strengthens the idea that one (of the Achronim) cannot use the Midrash as a Halachic source based only on his conception.
The Midrash, after it's composition, was a great influence on later Midrashim e.g. Shois Raba, Bamidbar Raba and Dvarim Raba.