Are there samples of false but beneficial beliefs? Many people have false beliefs.

The fact that they have those shows that those beliefs may carry some evolutionary advantage.

If so, how?

How can false beliefs be profitable?
I guess it depends on what you mean by "false beliefs". If you mean beliefs such as magical thinking, illusory patterns, superstitious beliefs, etc, then the following may help. 

From an evolutionary perspective, Foster and Koko (2009) argue that false beliefs such as superstitious thoughts/rituals may arise due to the consequence of making a false-negative error (i.e. dismissing a real pattern) being potentially dire, while making a false-positive error (identifying a non-existant pattern) may have little consequence at all.

A psychological explanation that is popular is the formation of false-beliefs as a means of gaining control over ones environment. Keinan (2002) proposes three ways in which engaging in false-beliefs may improve one's sense of control:


these beliefs may provide an explanatory power over perhaps
inexplicable events, making the environment seem predictable and
controllable;
false-beliefs may appear to provide a means of controlling a situation, such as engaging in a superstitious ritual;
engaging in false-beliefs may create a self-fulfilling prophecy (the person's faith in their beliefs may make them more optimistic of the situation, reduce their stress, and subsequently improve their performance).