Block on Trump's Asylum Ban Upheld by Supreme Court
Yes. Twitter and social media can be official public forums with constitutional protections when used for official government communications. That's not to say that Twitter or Facebook are themselves public forums, rather these sites provide a space for public forums to be held.
Like consumers fidgeting with most emerging technologies, the law often seems confused with how to handle new and even old tech. Recently, questions abound whether a government official can block a citizen from an official communication channel, especially when an official's personal account is used as the official communication channel. The president, and other members of government, are currently finding themselves defending lawsuits over this very issue, likely due to President Trump's extensive use of Twitter.
The Social Media Contract
The internet platforms are by no means traditional public forums. The companies that run social media websites are private and can legally censor and remove publicly, and privately, posted content. They can also ban, suspend, or limit users, for pretty much any reason. Additionally, social media users can narrow their privacy settings to block individuals, and even specific words, from showing up in their media feeds.
But, nevertheless, the very nature of governmental use of social media lends itself to being the most public of public forums. When government officials use social media channels to make official communications, there is a developing line of cases that establish these channels as limited public forums. Essentially when a government entity or official creates a social media channel for official communications, it converts the channel from a private forum, to a limited public forum, which entails not-so-limited First Amendment protections for those who participate in the forum.
Restricting Social Media Speech and Access
A primary cause for concern is that social media can often elicit controversial and emotional discourse, which can quickly devolve into name calling, shaming, meme-ing, and much worse. As such, it makes sense that a government official or department would be able to restrict speech, or even access, to some individuals in the public forum. However, doing so requires careful thought and application of content neutral restrictions in order to not violate the First Amendment.
For example, a constituent yelling obscenities into the microphone at a public town hall meeting, or on social media, is likely to be removed. However, an unpopular opinion posted on a city government's social media page in response to a new city project cannot be deleted or censored without violating the First Amendment, unless it violates content neutral restrictions. In most cases, of course, this is pretty obvious stuff.