In the US, it is not illegal to lie in general. This includes lying about someone: it's not illegal per se to lie about them. What is illegal is slander and libel: lying about someone in a way that hurts their reputation. The defamation doctrine in the US is generally a common-law doctrine (i.e. the rules and limits are based on court decisions, rather than on laws passed by legislatures), although it may differ state-by-state. Depending on the state, some defamation may be criminal; there is no federal criminal defamation.
US defamation law is largely defined through its interaction with the First Amendment. While libel is not constitutionally protected, punishment for libel is seriously limited by the need to avoid either punishing protected speech, or chilling potential protected speech (i.e. discouraging people from saying something that would in fact be protected, because they aren't sure whether or not it's protected).
Libel in the US only applies to a false statement of fact, or an opinion which implies some false fact. If it can't actually be proven incorrect, it can't be libelous in the US. The question of whether it's a statement of fact doesn't just depend on the literal speech; it includes things like the context, and is a question about what a reasonable person would think. If I were to claim that someone was "literally Hitler," for instance, no reasonable person would think I was seriously claiming that the person was the former leader of Nazi Germany.
Now, no reasonable person who is familiar with Twitter would ever assume that the tweet meant Obama literally stood up in front of the UN and said "Please accept this nothingburger in place of a respectable climate plan." So, it only counts as libel if a reasonable person would think it implies some fact. But a reasonable person familiar with Twitter would most likely think Miesel is saying "The president's pollution plan is a pointless piece of political puffery planned to placate principalities and potentates." This is basically a matter of opinion.
Even to the extent that it's not a matter of opinion, public figures in the US cannot win a defamation suit unless they show "actual malice:" the speaker must actually know or actually strongly suspect that their statement is false in some material way. It's not enough that a reasonable person would think "this might not be true;" the speaker themselves must doubt the truth of it (they must be reckless, not just negligent). Courts are also extremely deferential to defendants in these cases. While it is technically possible for a public figure to prove defamation, it is exceptionally difficult. If the person didn't know they were falsely attributing the quote, and honesty thought it was correct, they're in the clear. If the quote isn't supposed to be a statement of fact, but it implies false facts, but the speaker honestly thinks those facts are true, they're in the clear.
Private figures don't have to meet the actual malice standard to prove defamation. They still need to show that the statement is a statement of fact or something implying false facts; if it's obviously a summary of something they really said, possibly with added editorial comment, they can't prove defamation.