Yes, art censorship is a big problem in China, because that is one way that they can control public opinion. Art is often a reflection of person opinion on current events. Art censorship is similar to the way that they control the Internet in China. They have to have control so that people don't think for themselves.
China has a massive amount of censorship and art is not excluded from this. Foremost the issue of anything political being the most censored and tightly controlled topic in China. While it's unlikely that a landscape artist painting a cityscape of Beijing would, in general, find themselves censored - they might find such censorship if they decided to paint Beijing with a focus on its massive smog problem, which is a substantial political topic in China. So, in short, art censorship is a big problem because it limits political free speech.
Ai Weiwei is just one of the many artists who have been censored, harangued, arrested, exiled, or otherwise had their artistic careers derailed by an overzealous government bent on restricting free speech and free artistic expression by the average person. Though inroads have been made since China's increasing acceptance of Capitalism and foreign investment, which bolstered the art-buying and patron class, Chinese artists are not yet free to create the art they wish.
The Chinese Government is controlling of the media and art expressed by it's citizens but it's not to an oppressing level, I believe over time the Chinese Government will gradually relax its censorship of certain art on controversial areas and it won't be a problem at all, so I don't see this as a pressing issue.
While there is a lot of censorship, the arts in China do not seem to have suffered. Traditional art, that is--like painting and pottery and sculpting. Perhaps internet based art has suffered at the censorship in China, but that is just me guessing and making a possible assumption. The arts seem to be doing well overall.