Morning Star Loc: West coast, North of the 49th N.

Not necessarily always. I have taken many photos with the lenshood on and using the on-board flash, and without getting the dreaded shadow along the bottom edge of my photo.
It all depends on how close you are to the subject and what focal length you use.

When I have had the external flash on the hotshoe, I have never had a shadow on any photo, with or without lenshood.
I can see where this might happen with macro photography, but I "don't do" macro.

amfoto1 Loc: San Jose, Calif. USA

Not necessarily....

The best solution is to simply forget about using the built-in flash. It's anemic, slow to recycle, draws power from the camera battery drastically reducing the number of shots per charge, and is located in the worst possible place for ugly shadow and redeye effects.

A much better solution is a separate, accessory flash on a flash bracket, attached to the camera's hot shoe via an off-camera shoe cord. It's more powerful, often is faster recycling, has it's own power source, and since it's off to the side and higher above the camera - away from the lens axis - less likely to cause redeye and in most cases will throw shadows down and behind the subject.

An accessory flash on a flash bracket also won't have problems with leaving the hood on the lens.

A hood prevents oblique light from striking the lens. It also can be helpful preventing physical bumps against the front of the lens. Better than a "protective" filter. But if using a filter - and maybe a little ironically - it's even more important to use a lens hood since a filter introduces at least two more glass-to-air boundaries for light to cross, increasing the chances of flare or other effects from oblique light.

GaryS1964 Loc: Northern California

In my recommendation to remove the hood when using flash I said, "may" cause a shadow. If you are using flash you probably don't need a lens hood so why not just remove it to be certain you won't get a shadow.

When I was at Disneyland I was there from 10am to midnight carrying only my camera on a sling strap. Had the lens hood on during the day and just left it on when it got dark without thinking about it. I got the dreaded shadow when I took some closeups. Now when it gets dark I take it off and put it in the locker.

As for not using the popup flash sometimes you have need for a flash while you are wandering around somewhere for hours but you don't want to have a regular flash attached to your camera all that time and you also don't want to be carrying a backpack. Disneyland is an example.

dsmeltz Loc: Philadelphia

Since almost everything else has been well covered here, I will limit myself to one comment on a topic that I think (and I appologize if this was already covered) was missed. With some lens hoods on some zoom lenses you have to reverse or remove the lens hood or it might end up in your picture. This is when you are at the widest end of the zoom. But it does not happen often.

hochspeyer Loc: Chicago

I don't think this has been mentioned, but a very practical aspect of protection (at least for me) is the inch or so of it puts between my lens and things that could harm or dirty it... tree branches come to mind.

john clayton

Crop sensors are less likely to show hood shadow as full frame cameras. This is due to the lesser A O V. By the way I noticed a silly thing when trying a lens hood on my Tamron 18-270 lens. Users complained about the stiffness on the turning of the zoom ring. If you hold the zoom behind the lens hood between finger and thumb you can push or pull the zoom. Very smooth action.