Is a private pilot license holder in the USA who is giving a scenic tour with friends allowed to let them fly the plane? Are there any restrictions on when this can be done?
In my mind, this could be done safely while at altitude and the pilot could let them do a few shallow turns. They wouldn't want them doing anything complex and would want them to instantly let go if asked.
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.
AOPA has an article under Pilot Resources that specifically covers this situation called Logging Pilot-in-Command (PIC) Time (emphasis added by me):
Unlike driving cars, the PIC may allow anyone, including a non-pilot, a pilot who may not legally act as pilot in command, or another fully qualified pilot fly the airplane, or be "sole manipulator of the controls" during the flight. The PIC is not required to sit in the left pilot seat. Regardless of where the PIC is sitting in the airplane or who is manipulating the controls, the PIC is ultimately responsible and accountable for the safety and operation of the flight.
If you are the PIC letting someone else fly, remember (like they say above) that you are still responsible for everything that they do, so make sure that you never put yourself in a situation where you are not able to maintain (or regain, if necessary) control of the aircraft.
I would recommend that you carefully brief the passenger about not fighting you on the controls and to immediately let go if you tell them to. Do this before you let them fly and be clear that it is about flight safety. Most of them will be listening carefully once you say that. :-)
I have done this plenty of times, and another thing you can do (especially when at lower altitudes) is to keep your hands and feet on the controls at all times so that you can stop them from doing anything too out there.
Now that being said, air carrier regulations specifically prohibit pilots from letting passengers fly so the above would only apply to private flights:
§135.115 Manipulation of controls.
No pilot in command may allow any person to manipulate the flight controls of an aircraft during flight conducted under this part, nor may any person manipulate the controls during such flight unless that person is—
(a) A pilot employed by the certificate holder and qualified in the aircraft; or
(b) An authorized safety representative of the Administrator who has the permission of the pilot in command, is qualified in the aircraft, and is checking flight operations.
§121.545 Manipulation of controls.
No pilot in command may allow any person to manipulate the controls of an aircraft during flight nor may any person manipulate the controls during flight unless that person is—
(a) A qualified pilot of the certificate holder operating that aircraft.
(b) An authorized pilot safety representative of the Administrator or of the National Transportation Safety Board who has the permission of the pilot in command, is qualified in the aircraft, and is checking flight operations; or
(c) A pilot of another certificate holder who has the permission of the pilot in command, is qualified in the aircraft, and is authorized by the certificate holder operating the aircraft.
