A fake post in the sense of one that appears to be a Twitter or Facebook post (or a post on some other well-known social media site) but is in fact nothing of the sort, might be an infringement of trademark rights. Presumably such amn imitation would display the logo and other trademarks of the site being imitated, without permission, and in a way likely to confuse readers about the origin and affiliation of the display. Whether this would be considered to be "used in trade" or would be otherwise subject to a suit for trademark infringement is hard to say.
A fake post in the sense of one actually posted to a social media site, but under an assumed name, possibly under a name that seems to be associated with a real person, but in fact is not so associated, probably violates the sites TOS. That would be a breech of contract, but such breeches are, I understand, rarely pursued in the form of actual lawsuits.
If a fake post in either sense uses the name or likeness of a real person without permission, that might violate the personality rights of the person imitated. Such rights vary considerably by jurisdiction. In some jurisdictions they apply only to the use in marketing. In some jurisdictions no such rights are recognized. If a fake post attributes to a real person statements which that person did not in fact make, and if those statements are offensive, or harm the person's reputation, there might be a case for defamation. If a fake post reveals, or purports to reveal, private information about a real purpose, in some jurisdictions there might be a tort case for invasion of privacy.
If a fake post attempts to deceive people to the materiel benefit of the poster, there might be a case for fraud.
If the fake post serves to advertise some product or service, and if the false name conceals that it was made by or on behalf of the seller or maker of the product or service, or falsely appears to be from some reliable source, such as an expert or a celebrity, then under US law that might be an unfair trade practice. The US FTC can take action in cases of unfair trade practices, if it sees fit.
In the US at least, simply posting under a false name is not a crime, and indeed may well be protected speech under the first Amendment to the US Federal Constitution. See this question and its answer for more details.