How do the LCD projectors generally work compared to the overscanning of TV sets?

Do they have a similar effect or do they display more of the incoming picture ?

LCD and other pixel oriented displays, map and display by default the full image content. This is allways the case for computer images, and mostly true for video. One can of course allways suppress (blanking) the outer area's and futher fill the projected image using extra zooming or throw distance change, if "overscan" is needed. Large overscan (10%), like it was used in the past(and still is used to a lesser degree in TV) in order to compensate for ageing, mains voltage changes, signal tolerances...in the old days, is no longer existing. So, don't believe the people telling you that you will not see vignetting on yr TV or projector (even CRT) because of applied "overscan"...better verify first!

I have had mixed results. Usually you get the whole picture, but there are some projectors that act like a TV and crop your image. Some you can go in and keep them from cropping, some you can move around the crop area,
but the image is gonna get cropped.

My advice, if you are planning on renting a projector for a show, go to the
rental house well in advance and get a demo to make sure. Then
reserve THAT projector for your date(s).

PS Also ask them what happens if the bulb burns out? ;)

I created a 10 minute video which I projected in front of 250 people via a digital projector. As I was creating the video I was worrying about the cropping that my camcorder was doing and wondering if the same amount of croppage would be shown when I projected it. I didn't have time to test it, so I just hoped for the best. After creating the whole video and showing it on the digital projector I learned the hard way that my projector did not crop my video. I had more showing than I wanted and some stuff was seen that wasn't supposed to be. I think that I was the only one who noticed it though. The average person doesn't really look for/see those kind of things. So, to answer your question, the projector I used showed all my video and did not crop it (unlike a TV). But I am sure that different projectors vary, so, if possible, you should do what Jacques said and test your projector before you project! Good luck!

I'm actually finishing a project that will mostly be shown via different LCD-projectors (so testing one is not useful), and sometimes (rarely) on a TV set.

To me it's more important that everything is visible, not vice versa. That's why it seems that the smartest thing to do would be rendering the project with somewhat smaller picture than the full resolution would allow (adding some black on the sides).

What I might lose is some 40-50 pixels (PAL) in case of a 100% projector but on the other hand gain full control of the visual impression in case of a cropping device.

Whichever I choose might lead to unpleasant results. Still I'd rather choose the one that gives me full picture every time.

Good idea. You could also make several versions. One regular, one
slightly smaller image surrounded by black boarders, and another
even smaller image with large black boarders.

It is amazing how each TV/projector has its own crop factor.