The original phrase "falsely shouting fire in a theatre and causing a panic" came from the opnion by Justice Holmes in Schenck v. United States, 249 U.S. 47 (1919) where it was a metaphor for speech that might validly be prohibited under he First Amendment. It served to introduce the famous "Clear and present danger" test. The issue in that case was whether distributing fliers urging people not to submit to the draft could be the reason for a conviction under the Espionage Act, or whether such action was protected by the First Amendment. Holmes held it was not protected. The case was not about any actual shout of "fire".
The Court has generally held that true speech may not be prohibited, and for this reason some legal scholars argue that the "private facts" tort is unconstitutional in the US. But in many cases the Court has held that false speech is also protected, so truth is not the primary criterion.
In Brandenberg vs Ohio 395 U.S. 444 (1969) the "Clear and present danger" test was replaced by a rule that only speech "directed to inciting or producing imminent lawless action and is likely to incite or produce such action" may be prohibited under the first amendment, and this is now the test for any crime of "incitement". Under this test Schenck would hav gone the other way. This test says nothing about the truth of the speech, and a perfectly true statement may nevertheless be incitement if it seeks to arouse lawless action, and is likely to succeed.
Near vs Minnesota held that bare truth must be a defense in any prior restraint case (unlike truth published "with good motives and for justifiable ends" which the saturate at issue in Near allowed as a defense), but went on to greatly restrict any prior restraint, even if the content was false.
Defamation is really the primary situation in which falsity is a vital element of the offense, and so truth is a defense.
Falsity is an element of perjury also, and so truth would be an absolute defense there as well. Falsity is also an element of the common-law tort of fraud, and of most statutory versions of fraud, but in some of those a true but misleading statement can still be held fraudulent, so truth is not an absolute defense.
I cannot think of another such legal situation.
Updatew:
A comment asked:
What if I inform all of Joe's neighbours that Joe had been arrested for murder - when I fully know that he was completely exonerated and the real murderer is in jail right now. It's literally the truth, but completely misleading.
In that case the law takes you to have made by implication a statement that Joe is currently under suspicion for murder, or perhaps a statement that Joe is actually guilty of murder. As those statements are false, and apparently knowingly false, they can be the basis of a defamation action. There is a lot of caselaw on defamation by implication, but in general one is taken to have made statements that are clearly implied by the statements one did in fact make. I don't think this is very different in UK and US law, but I am not sure about the UK. Anyway truth is not a defense here because the implied statement is not true.
