The name of the river that sweats "oil and tar" is the Thames.
The reference to the Thames can be found in the third part of the poem, called "The Fire Sermon."
In this section of the poem, the poet talks about how summer has departed without leaving traces of its previous habitation.
The river bears no empty bottles, sandwich papers,
Silk handkerchiefs, cardboard boxes, cigarette ends
Or other testimony of summer nights.
While the poet is used to seeing the Thames littered with the accouterments of summer reveling, he sees little to rejoice in the current season of his life. In fact, the poet is in a decidedly morose mood:
By the waters of Leman I sat down and wept . . .
Interestingly, Eliot himself received psychotherapy treatment in Switzerland; it was at Lake Geneva (or Lake Leman) that he wrote much of "The Waste Land
The third section of Eliot's poem mingles the horrific with the sensual. For example:
White bodies naked on the low damp ground
And bones cast in a little low dry garret,
Rattled by the rat’s foot only, year to year.
Mr. Eugenides, the Smyrna merchant
Unshaven, with a pocket full of currants
C.i.f. London: documents at sight,
Asked me in demotic French
To luncheon at the Cannon Street Hotel
Followed by a weekend at the Metropole.
The dead corpses and same-sex trysts were understood in Eliot's time to symbolize the degeneracy of modern culture. Thus, the river that sweats "oil and tar" is a symbol, an emblem of a culture that has lost its bearings.