A falsehood is (1) a lie, (2) an untrue statement, and (3) the practice of lying. Falseness is the quality or condition of being untrue. So a falsehood is something that has falseness.
Falsity is a variant of both words, though it’s more often used as a synonym of falseness. In fact, though falseness is far older than falsity, the latter is more common in web-searchable news publications from the last few years.
Falsehood is usually synonymous with lie or untruth—for example:
They discovered that in the crisis, Twitter crowds reflexively sorted facts from falsehoods. [Wall Street Journal]
The portrayal of protesters as dirty, crazed malcontents is a dusty old cliché based on a falsehood. [letter to Edmonton Journal]
Also note Perry’s casual falsehood about Obama supposedly proposing that veterans pay $200 before getting any health benefits. [Washington Post's The Plum Line]
Falsity is usually an antonym of truth—for example:
Full transparency would give a “proper benchmark” against which the truth or falsity of that charge could be assessed. [Financial Times]
Cervantes is not parodying the tales of chivalry but rather the inability to suspend the judgment of truth and falsity that reduces all narrative to one standard. [NYT Opinionator]
And falseness is a less common synonym of falsity—for example:
I find it to be a corrupt cesspool of disingenuous skullduggery, back-knifing and pretentious falseness. [Red State]
Harbach’s generosity is both that, and a lesson in the falseness of appearances. [Seattle Times]