Yep, totally true. Attested by multiple independent sources. The Bouphonia ("ox-killing") was a very ancient festival held annually at Athens from time immemorial to the Roman period.

The earliest reference to the Bouphonia is the Classical comedian Aristophanes, who has the character identified as "Unjust Argument" paint it as an old-fashioned, archaic ritual:

Just Argument: I will, therefore, describe the ancient system of education, how it was ordered, when I flourished in the advocacy of justice, and temperance was the fashion. (...)

Unjust Argument: Yeah, antiquated and Dipolian and full of grasshoppers, and of Kekydes, and of the Bouphonia!

-- Clouds 961-985

Various later sources explain for us what the festival entailed. Most briefly, we find a sentence in Aelian's Miscellaneous Histories 8.3:

This is an Athenian custom in which an ox is killed, but they declare that each of the people involved is innocent of murder, but they sentence the cleaver, and say it killed him. And they call the day on which they do this the Dipolia and the Bouphonia.

We get more detail from the travel writer Pausanias, who includes it in his description of the temples and altars found on the Athenian Akropolis (1.24.4):

Upon the altar of Zeus Polieus they place barley mixed with wheat and leave it unguarded. The ox, which they keep already prepared for sacrifice, goes to the altar and partakes of the grain. One of the priests they call the ox-slayer, who kills the ox and then, casting aside the axe here according to the ritual runs away. The others bring the axe to trial, as though they know not the man who did the deed.

Pausanias also helpfully explains the general Athenian habit of taking inanimate objects to trial, of which the Bouphonia is a ritual example. He claims that the origin of the Bouphonia was also the origin of the trial of objects for polluting crimes (1.28.10):

The Court in the Prytaneum, as it is called, where they try iron and all similar inanimate things, had its origin, I believe, in the following incident. It was when Erechtheus was king of Athens that the ox-slayer first killed an ox at the altar of Zeus Polieus. Leaving the axe where it lay he went out of the land into exile, and the axe was forthwith tried and acquitted, and the trial has been repeated year by year down to the present.

But the most detailed account occurs in Porphyry's long essay about why it is bad to kill and eat animals (On Abstinence from Animal Food 2.29-30). He gives a different backstory than Pausanias, involving a farmer killing an oxen in rage but later seeking atonement after a drought struck the city. He then offers a description of the festival itself:

They selected virgins who were drawers of water; but these brought water for the purpose of sharpening an axe and a knife. And these being sharpened, one person gave the axe, another struck with it the ox, and a third person cut the throat of the ox. But after this, having excoriated the animal, all that were present ate of its flesh. These things therefore being performed, they sewed up the hide of the ox, and having stuffed it with straw, raised it upright in the same form which it had when alive, and yoked it to a plough, as if it was about to work with it.

Instituting also a judicial process, respecting the slaughter of the ox, they cited all those who were partakers of the deed, to defend their conduct. But as the drawers of water accused those who sharpened the axe and the knife, as more culpable than themselves, and those who sharpened these instruments accused him who gave the axe, and he accused him who cut the throat of the ox, and this last person accused the knife; hence, as the knife could not speak, they condemned it as the cause of the slaughter.

From that time also, even till now, during the festival sacred to Jupiter, in the Akropolis, at Athens, the sacrifice of an ox is performed after the same manner. For, placing cakes on a brazen table, they drive oxen round it, and the ox that tastes of the cakes that are distributed on the table, is slain. The family likewise of those who perform this, still remains. And all those, indeed, who derive their origin from Sopater are called boutupoi [ox-killers]; but those who are descended from him that drove the ox round the table, are called kentriadai [goad-men]. And those who originate from him that cut the throat of the ox, are denominated daitroi [dividers] on account of the banquet which takes place from the distribution of flesh. But when they have filled the hide, and the judicial process is ended, they throw the knife into the sea.

What is all this nonsense about? The Greeks themselves (including Porphyry above all) believed that there had been a time when there were no animal sacrifices, and it was believed that killing animals was wrong. Therefore the first introduction of animal sacrifice had to be surrounded with elaborate rituals to absolve the killer of guilt. This is probably retroactive myth-making, but it highlights again that the Bouphonia was regarded as a very ancient ritual of which the origins lay in Athens' earliest myth.

The theatre surrounding the judgment on the axe or cleaver is more likely to have to do with the Greek notion that particularly heinous crimes, like murder, brought a pollution on the murderer that clung to their surroundings. This was why murderers were normally put to death or sent into exile; they weren't just supposed to be punished, but wholly removed from the community, to appease the gods. The ritual of the Bouphonia is designed to ensure that no human is actually guilty of the act of killing, or even of choosing which ox is to be killed (they select themselves by eating the grain first). When all have testified that the axe was the real murderer, the object could either be acquitted (according to Pausanias) or thrown into the sea (according to Porphyry). That way, the community was absolved of guilt and associated pollution. Presumably this ritual was to some extent supposed to absolve the Athenians symbolically of guilt for all animal sacrifices that year, since there were many other festivals in which goats, sheep, oxen and other animals were ritually killed.

I don't know where the Tumblr author got the idea that the Ancient Greeks were a rational people; they were anything but. The study of their superstitious and illogical habits begins with the still widely read The Greeks and the Irrational (1951) by E.R. Dodds; see also the more recent Rationality in Greek Thought (1996) by Michael Frede and Gisela Striker.