Shrek has unfortunate character design, especially for an animated movie. This affects the overall appeal of the image and of the animation. Basically... no matter how you pose those things, they're not going to look good. Shrek feels more like a struggle to get the models to look OK, and the struggle messes up the animation. When the characters look sexy from any angle, animating is easier and there is a constant level of quality and appeal all throughout. In Shrek, they look more like generic 3D creations... like some sort of Poser characters with better texturing and shading. Shrek tries to be cartoony-photoreal and it just looks weird and wrong. So to sum it up, bad design, leading to a lot of bad poses.
Shrek doesn't have bad animation, but I think the animation doesn't fit very well the character design. The design is more towards an awkward sort of realism, but the animation is a bit too cartoony, springy, bouncy, for this design. It just feels like the weight is off, because the weight is messed up with all the cartooniness happening. I may be wrong about this one though, since cartoon animation has been used successfully on semi-realistic humans. But there are shots in Shrek that just have this weird weightless feel...
Hands in the air. Too much of it. There are a lot of empty hand gestures in Shrek, hands hanging in mid air, looking IKish (with the elbows suspended too high, too un-gravitationally...). This is an animation problem, but it starts from story-telling and directing. A lot of time the characters don't have anything to do, in particular, and especially with their hands. So... instead of just doing nothing, or something meaningful, they float around... (Look at Merlin's hands in the image above for example. That's no way to hold a spoon or, especially, a 'plate'.)
The stiffness of Shrek may also be a matter of insufficient dynamics in terms of cloth and deformations and environment, effects, plants... The sort-of-realistic look of Shrek could use, I guess, more complexity in this area. It does have a ton of cloth simulation, but it doesn't look good or realistic, or complex enough, rather simplified. Overall, it feels stiff. I think that due to sheer complexity of the project, a lot of corners have been cut and lots of things have been automated and simplified. Which leads me to
Lots of ugly looking crowds and secondary characters. The creators of Shrek have used a procedural approach for generating the tons of characters in the movie, and despite the variety... they all look awful. Well, in fact, if the main characters look bad, the crowds couldn't look much better, could they... hehe. I think, again, the design is not helping this proceduralism at all. The procedural humans created for The Incredibles look really good (despite the fact that the creators of The Incredibles seem to have been not so pleased with the procedural-man approach to character creation). Anyway, different design altogether.
The facial rig, which in its day was probably awesome (I guess they've reworked a lot of it meanwhile, but all in all, the results are very similar), is not really that great for this design. It's supposed to be based on anatomical studies and all, but it's only a crude simplification of real human facial deformations. That being said, I'm sure it's a fantastic rig, for characters like
Shoulders. Square. Ugly. Most characters have these square... awkward... tubes... instead of upper arms and shoulders. I think they're among the weakest parts of both character design and 3D work - they also deform so badly, I think, because they've been set up to be friendly with the cloth simulation that has well known problems around areas like shoulders. 2 more tubes in the image below:
PS - I like this guy in the image below: