I understand Armstrong landed the Eagle manually, and Apollo 13 did a manual burn. Other than emergencies like that, did the Apollo astronauts do much piloting of the spacecraft.
All six successful landings were done manually, with the commander taking the controls at about 500 feet altitude. It was possible for the computer to complete the landing automatically, and Jim Lovell claimed that he was going to attempt that mode on Apollo 13, but all the commanders felt more comfortable in manual control in the final phase of landing.
It's worth noting that there were a number of different "manual" control modes. The one used for the final portion of each landing, control program P66, kept the computer in control of the throttle to maintain a selected descent rate, but letting the commander control the attitude (orientation) of the spacecraft and adjust the descent rate. Digital Apollo is a great source for more information on this, as is MIT's report on the Apollo Lunar Descent Guidance system.
Apart from that, all the maneuvers where multiple spacecraft were operating in close proximity were flown manually:
The “transposition, docking, and extraction” maneuver, where the command module turns around to retrieve the LM from the S-IVB third stage was flown by the command module pilot.
Prior to landing, the CSM and LM would separate and the LM would turn around slowly to allow the command module pilot to inspect the LM visually. This maneuver on Apollo 11 is shown in this video.
After the LM ascended from the lunar surface, the initial part of the rendezvous with the CSM was performed by the LM under computer control, but the final approach and docking was again flown manually by the command module pilot.
The re-entry could be flown under computer control or manually; by rolling the command module to control the direction of lift, the trajectory could be flown more steeply or shallowly. I don’t know off the top of my head if the manual option was ever used here, or how often.