Example of species that went extinct because of excessive hunting by non-humanoid predators?
The Brown Tree snake (Boiga irregularis) is responsible for five extinctions on the island of Guam. Some of these species existed only on Guam, while others exist elsewhere and so it is the Guam subspecies that has been lost.

Guam flycatcher (Myiagra freycineti). Last sighted in 1983. 

Guam rail (Gallirallus owstoni) is extinct in the wild, but is now bred in captivity and has been introduced to the island of Rota.

Guam kingfisher (Todiramphus cinnamominus) is also extinct in the wild and exists only in a captive breeding program.

Guam flying fox (Pteropus tokudae). Went extinct in the late 1960's possibly as a result of the Brown Tree snake.

I had a suprisingly difficult time obtaining accurate lists of the Guam extinctions.

For an example of overgrazing leading to a local extinction, see St. Matthew island in the Bering Sea. The US Coast Guard introduced reindeer in 1944. By 1963 there were ~6,000. Without any natural predation to control the population, they are considered to have overgrazed, causing a population collapse to 42 by 1965 and no reindeer population by the 1980's.