Does the engine have to be running in order to turn the steering wheel on a boat with hydraulic steering, either full hydraulic or the add-on system w/o the hydraulic helm?

Not on mine

You can steer with either system with the engine off. It may require 2 hands though cause it's stiff.

I had to maneuver into my slip last season on 1 engine and as luck would have it the p.s. pump was on the blown motor. I still managed to get that in but man was it stiff!

Full hydraulic should allow you to steer minus the power assist. Add-on type is a different story, the cylinders outside are not getting any fluid from the valve inside and the cable alone won't override them unless they are real weaklings. Usually this would require the unfastening of the hydraulic lines to allow the cable to do its job.
Just my .02

I can't turn my MM add on system more than and inch or two without the engine running.

I have add on MM. I ran one motor out of gas last summer (3 am) and it was the one with the pump. I could hardly turn the wheel. By turning the motor over with the starter the pressure was released and I could turn it for course corrections. I was able to stay up one one.

I have full M&M hyraulic steering. It can be turned without the engine running but man you've got to put some umphh into it.

I stand corrected. I have the full helm and I can steer mine.

Latham - Pretty hard to steer without the engine running.

BTW - it's ALWAYS gonna be the engine with the pump that has problems -