This NASA website states
"NASA is unlikely to find any use for the L3 point since it remains hidden behind the Sun at all times. The idea of a hidden "Planet-X" at the L3 point has been a popular topic in science fiction writing. The instability of Planet X's orbit (on a time scale of 150 years) didn't stop Hollywood from turning out classics like The Man from Planet X."
where "The Man from Planet X" links to IMDB: The Man from Planet X (1951). Since Planet X supposedly passes by the earth after leaving it's own sun, this should probably point to Journey to the Far Side of the Sun (1969) instead.
For the sake of discussion of the question of stability: if there hypothetically were a planet, with say a mass roughly equivalent to earth, then would it really be unstable in the same way that a low-mass object would be unstable at Earth's L3? And wouldn't that mean that both it and earth would be similarly unstable? Or does this only refer to the idea that it could't remain hidden behind the sun for hundreds of years, as a very old calculation summarized on this webpage which I found here seems to suggest. Has this calculation been repeated and reported more recently?
note: this all stared from a conversation below this question
note 2: while this question is about the statement about stability of a planet opposite earth, from this answer it seems that the quote above should point to Journey to the Far Side of the Sun (1969) and not point to The Man from Planet X (1951)