The exception that proves the rule is a good example. According to Wikipedia, based on Fowler’s Modern English Usage, the phrase has its origin in Roman legal doctrine, and at full length reads:
Exceptio probat regulam in casibus non exceptis or
The exception proves the rule in cases not excepted.
For instance, though not matter for a major legal case, suppose you see a sign at a stately home saying “sorry, we’re closed to the public today for a special event.” You would take this as an exception proving the rule that the stately home is generally open to the public.
Fowler’s example quoted in Wikipedia has more legal flavour:
Special leave is given for men to be out of barracks tonight till 11.00 p.m.; "The exception proves the rule" means that this special leave implies a rule requiring men, except when an exception is made, to be in earlier.
Current usage is more along the lines: “I always keep my New Year resolutions.” “Well, last year you said you would quit smoking, and you didn’t.” “Oh, that’s the exception that proves the rule.” This is obvious nonsense: this "exception" disproves the rule. Even if the claim had only been "I generally keep my New Year resolutions," an "exception" would at best not refute the rule, but never prove it.
So, in the original meaning, you realise something is an exception, and from the exception you infer there is a rule. The exception is therefore a case to which the rule does not apply. In current usage the exception is a violation of a rule previously stated. And then people nonsensically use the phrase to dismiss the contradictory evidence. In the worst cases people may even believe that a rule is proved true when found violated.
The exception that proves the rule is the topic of this question.