Are pilots allowed to let passengers fly the plane?
For a non-commercial flight under Part 91 rules there are no issues I'm aware of with letting a non-rated passenger handle the controls - and it's done quite frequently (I can't think of any pilots I know offhand who haven't had someone else take the controls for one reason or another).  

The rated pilot is still pilot-in-command of the flight and ultimately responsible for everything that happens, so they need to be relatively certain their friend will let go of the controls if told to do so, and they need to maintain situational awareness (traffic scan, airspace, etc. -- if the friend blunders into Class B airspace it's going to land in the pilot's lap) and be ready to take over if the friend starts doing something that could jeopardize the safety of the flight.



There are some common-sense limits here: something insane like have two non-rated friends at the controls the rated pilot being PIC from the back seat giving them instructions over the intercom would be a no-go.
Similarly while this sort of thing is a total non-event in an aircraft with typical dual controls you would probably want to have a higher level of trust in your friend if you're flying something with a throw-over yoke.