Yes, topics related to Barack Obama are often censored if they do not cast Obama in a positive light, because social media websites need to make sure that they keep Obama happy. Even Facebook has censored conservative groups and comments. Someone published a bunch of editorials under the name "Ellie Light." They were designed to cast Obama in a positive light. The same is true with social media.
The American press is very cozy with Obama. In the last presidential race, no reporters asked him about the promises he has broken such as the failure to close Guantanamo Bay, immigration reform, and the failure to prosecute anyone for the housing crisis. The only reporter to even call him out on any of his broken 2008 promises was Jorge Ramos from Telemundo, a Spanish Network. CNN, MSBNC, ABC, and CBS pretty much just asked him questions that were pre-approved by the White House. If anyone dares criticize the work he has done, figureheads such as Al Sharpton of MSNBC are quick to point out that it's racism. Even Liberals have a hard time criticizing him for the same reason that Conservatives cannot. Any criticism is met with accusations of racism.
It has been a national disgrace as to how President Obama has been treated. Trying to deny him his birthright for instance. Never giving him his due when he does make things better for the U.S. I have been sad to see how many negative things have been said about both his family and himself that has nothing to do with Politics. It seems, like Bill Clinton, that for whatever reason, President Obama was not going to be treated as he should.
No, topics related to Barack Obama are not prohibited for no reason on social media sites, unless they cast him in a favorable light. I see all kinds of post on social media that are very critical of the president, some are downright hateful. In fact I think he is criticized even more than Bush was on social media.
It seems lately that anything related to Barack Obama is automaticly taken and turned into an evil vicious plot by the extreme parts of the right wing. There is plenty of liberal spin surrounding the guy for sure, but I think there's enough media choices that all sides of Barrack Obama, particularly the negative, are explored.
If you scroll through Facebook posts and Tweets on any given day you will see tons of comments on topics related to President Obama. There are many many unfavorable comments, posts and memes which are negative, racist, hateful and disrespectful. For the most part, those types of comments are not censored and they are certainly not prohibited on social media sites.
We have the freedom to speak our minds in the United States. As such, topics related to Barack Obama aren't seemingly prohibitive. People criticize the president all the time. The difficulty is that most people who use social media aren't the ones who voted against him. Many of the people who voted for conservatives are older white men who don't have Twitter or Facebook accounts.
I have a diverse group of friends so I do get a fairly cross representation of what is out there on social media. There is a lot of pro-Obama stuff, but a lot of it comes from sites I have "liked". Obama is a likable person, it is hard to attack him or his family, but I do get a lot of anti-gun legislation stuff and opinions that are not favorable to Obama. I have noticed that Fox News will hardly ever report a good thing about Obama, but they are not social media. I have also seen downright nasty stuff posted about Obama...so I really don't see any bias.
Now, I don't know about all social networks, but I am friends with both Democrats and Republicans on Twitter and Facebook and from their posts there is no censorship on those forums. When Obamacare passed I saw plenty of posts both in favor and against the reform and many of them either thanked or hated President Obama.
Articles related to President Obama are all over the Internet whether they cast him in a favorable light or not. It does depend on the news source as to what context they choose to present the article in, however. Fox News is unabashedly conservative and will paint him as a socialist president out to squash the American government system, while MSNBC, a liberal news source, will present his policies as beneficial to the country. There are many groups on social media sites more than willing to present topics related to Barack Obama that are not cast in a favorable light. In fact, some groups exist solely to promote him in an unfavorable light.
I don't think that it is accurate to say that all social media sites are censoring anything that is negative is regards to President Obama and the job that he is doing. Maybe it is possible that simply many more people approve of his job, and as such don't post as much bad stuff.