Here's some help in the form of a partial list of rivers that form borders based on a quick google search and a whirl around google maps. This list is probably not complete:
China, Burma, Laos, Thailand, Cambodia, Vietnam (border the Mekong)
In South America:
Rio Uruguay: Argentina and Uruguay.
Courantyne-Koetari River: Guyana-Suriname.
Marouini-Lawa-Litani River: Suriname-French Guiana.
Rio Parana: part of the border between Argentina and Paraguay.
Rio Paraguay: part of the border between Paraguay and Brazil.
Rio Mamore: part of the border between Bolivia and Brazil.
Rio Putumayo: part of the border between Colombia and Ecuador.
Rio Napo: a very small part of the border between Ecuador and Peru.
Rio Orinoco: part of the Colombia-Venezuela border.
Rio Takutu: part of the Guyana-Brazil border.
In Europe the Danube borders several countries.
In Africa the Congo, and its tributary the Ubangi form partial borders with a few different countries.
(The ones in quotes I just copy-pastaed from other people's research.)