I utilized a couple of slang books. I suspect the name merely comes from the river's makeup/appearance, such as "swamp water", "tar water". The southern boys can probably tell us if that's the case.
Since the early 1700s North Carolina has been this nations largest producer (95%) of naval stores (tar, pitch, turpentine) that was vital to the construction and maintenance of wooden ships. http://www.northcarolinahistory.org/encyclopedia/103/entry/
The Tar River, which begins in my home county, and eventually flows into Pamlico Sound may have received this name because of its use in the transportation of naval stores. It has also been suggested that it was named by early British settlers after the River Taw in England. Joe Sweeny would have crossed the Tar River numerous times during his traveling days as a circus performer in the 1830s.
http://research.culturalequity.org/rc-b2/get-audio-detailed-recordi... ... of course it may be related in name only.