Just about any sort of non-professional animator will face problems of animating without much aid or funding.  It's not limited to LEGO.

One such problem we face, however, is that there are some pieces which simply do not exist, and it can be difficult to make a specific story since certain pieces you might need aren't anywhere.

I'd like to make many more Riigo-Faloo brickfilms, however, most of the characters have very particular faces and outfits which would require specially printed minifigures.  Comrigone wears a unique purple suit and a unique scarf, Nirim has a white face with a great big scar across it, and several other clowns have very special markings upon their faces.

It is possible to print new pieces or create new molds, however, they often fail to look consistent with everything else LEGO.  There are only a scarce few people who are capable of producing custom LEGO bricks and prints of acceptable quality, but none of them brickfilmers.

There is also the problem of facial animation being very hard to do.  Most people fail miserably, and preferably if you can't do it well you should just stick with a single expression with possibly one or two more faces to be changed out if you have them.

To deal with this problem, I try to voice my characters as though they are constantly stuck in a single emotional state, or possible two if there is a face change.  This way the single minded or polar opposite feelings of the character matches the face.
When I had Brandon do his lines for Welcome to Darkmoor, I instructed him to say everything as though he was raging mad because his character's face is always irreversibly angry.  Everyone else in Darkmoor speaks in a creepy happy way, always consistent with their doll like smiles.
I guess it's not very good to have such a limited and one-dimensional emotional range for one's characters, but I like keeping the facial expressions and feelings as consistent as I can.

Size. Lego is a lot smaller than most things used to animate. Thus making it harder to get close ups. easer to bump. And Lego reflects a lot thus making it harder to get good lighting.

As I've described before, I think some aspects of brickfilming are extremely niche. However, I also think brickfilming shares a lot more in common with other hobby stop-motion filmmaking than some think. I'm sure other animators have experienced set bumps, light flicker, and so one. I don't think it's correct to compare us to Aardman or Laika. Those are professional studios. Instead, look at the thousands of stop-motion films on YouTube that feature puppets or claymation or whatever. You'll see many of the same problems that brickfilmers face.

However, brickfilmers do have to deal with reflections on shiny bricks and minifig torsos. I hate that.

YouTube | Twitter | Brickfilm Archival Project
Brickfilms Wiki | BiM: The Documentary

I'd say scale is by far the biggest challenge to face when brickfilming. Occasionally, a large scale is fitting for the film, enhancing it in ways that are non-comparable with other animation forms. However, it can be incredibly difficult to shoot close-ups without expensive equipment. Add in the fact that you might have to occasionally tear down part of the set just to switch the shot, and you have a really complicated situation on your hands.

Facial animation, like Squid said, is also a tough one because LEGO figs can't make facial expressions. Unless one is a skilled digital artist (or one erases the face of minifigures), facial animation is a no-go for most brickfilmers.

Brickfilming does require talent and dedication.  Lots of it.  Just like any other type of animation.

Brickfilming doesn't require the artistic skill of claymation or hand-drawn animation. You need sculpting/drawing skills for that. With LEGO you just need a ton of bricks and a willingness to learn on the job. Brickfilming is one of the few hobbies you can just throw yourself into right away without any preparation (grab a camera, download some free software, get some LEGO and GO!) - which is why there are a lot of crappy animations online, because all those people have only just started. The majority of them have a description saying "Yeah, I was bored one day so I made this"

I would say thats a unique challenge, but with pretty much everything you do you're going to suck at first. You don't spring out of the womb fully formed and able to compose operas. You emerge naked, covered in gunk and unable to feed yourself or go to the toilet.

I try to make up for my complete lack of artistic skill (can't make music, can't draw, can't sculpt, can't really act, can't 3D model) by ensuring my writing is good. The reason why the majority of my Brickfilms are just people talking is because I'm best at scripting dialogue. Anyone who says thats boring has clearly never watched a play, or listened to a radio show, or watched a low-budget film/TV series, or gone outside. Anyone who doesn't care for hearing people talking has clearly never observed an interesting conversation in their life - real or scripted.

I suppose thats the challenge we face: taking a straightforward hobby and making it interesting and exciting. Anyone can make LEGO move - but can you make it tell a story?

I think one of the big issues I see with animating with LEGO that I don't have to deal with if I'm working with clay or some other non-LEGO animation style is organic shaped sets and props. I've always wanted to create a LEGO film with a Dr. Suese style setting but building houses that aren't conformed to right angles is very difficult with the LEGO system. It is defiantly possible to create things that are more organic looking. But you just don't see people creating films that are trying to push this aesthetic aspect in films.

A.B.S. Snorter and his Lego brick sorter by Legohaulic, on Flickr

Crow's Nest by Legonardo Davidy, on Flickr

When I got into stop motion (via some friends who used LEGO and a web camera) I wasn't aware of the existence of the AFOL community. A while after making our first LEGO film we discovered Brickfilms.com and realized how bad our 7fps animation was. It took a few more years to really see the amazing work that adults were doing on just MOCs and I really would like to see more attention to sets, and settings in brick filming in general.

I had never seen Weltraffer, thank you for that.