This is a good question. The information available is very complex, so I have gleaned the most relevent information to answer the question. To begin, @lasersauce made the correct observation that
He [Tammuz] appears to have been a god of the spring, and the myth regarding him told of his early death and of the descent of Istar, his bride, into the underworld in search of him.
In this regard, Becking and Dijkistra (1996) provide important historical details. That is, there are similarities with the death of the Canaanite God Baal and Tammuz written in the In the Ras Shamra texts. In one of the studies the writer observes that
The mourning for Baal as a vegetation deity in eclipse suggests the weeping for Tammuz, also a vegetation deity, by the woman of Jerusalem in the sixth month (Ezek 8:14) and, more directly, the public mourning for Hadad-Rimmon (the Canaanite Baal) in the valley of Megiddo (Zech 12:11). (1).
Additional sources are here, as well as this post.
In summary, the sources indicate several parallels between Tammuz and the Canaanite god Baal.
