The most (and maybe only) clearly recorded incident was in 206 BC when a novice vestal was whipped for letting the flame go out.
Several historians record an incident that occurred in 206 BCE, in
which a novice priestess had negligently allowed the sacred flame to
be extinguished during her shift. After she was accused of a
transgression and threatened with execution, the Chief Vestal Aemilia
miraculously rekindled the flame by applying a piece of her stola to
the cold coals in tandem with a supplication to Vesta.
Source: Joshua Michael Roberts, 'Rome's vestal virgins: public spectacle and society' (Masters Thesis, Western Washington University, 2012)
The 'several historians' are cited in a footnote:
Dionysius of Halicarnassus 2.68; Valerius Maximus 1.1; Livy 28.11;
and Propertius 4.11.53-4
Livy records this thus (context is during the 2nd Punic War):
The extinction of the fire in the temple of Vesta struck more terror
upon the minds of men than all the prodigies which were reported from
abroad, or seen at home; and the vestal, who had the guarding of it
for that night, was scourged by the command of Publius Licinius the
pontiff. Although this event was not appointed by the gods as a
portent, but had happened through human neglect, yet it was thought
proper that it should be expiated with victims of the larger sort, and
that a supplication should be made at the temple of Vesta.
This specific incident is mentioned in several other modern academic sources (for example, here and here), none of which cite any further such incidents. This does not, of course, mean that 206 BC was the only time that the flame went out.
Another possible incident of the flame going out is mentioned by Plutarch in his Life of Cicero. Referring to a speech Cicero was preparing which touched upon the trial in which Cicero's sister-in-law Fabia, a Vestal, and the senator Catiline were accused of having a relationship, Plutarch writes:
While Cicero was in this perplexity, a sign was given to the
women who were sacrificing. The altar, it seems, although the fire was
already thought to have gone out, sent forth from the ashes and burnt
bark upon it a great bright blaze.
There may well have been other occasions as Dionysius of Halicarnassus mentions that the flame going out was one of the signs of a Vestal breaking her vow of chastity:
There are many indications, it seems, when a priestess is not
performing her holy functions with purity, but the principal one is
the extinction of the fire, which the Romans dread above all
misfortunes...
Dionysius (69.2) mentions one case in which the extinction of the fire was relevant - or not in this case, as the fire did not go out:
They say that somebody unjustly accused one of the holy virgins, whose
name was Tuccia, and although he was unable to point to the extinction
of the fire as evidence, he advanced false arguments based on
plausible proofs and depositions...
Aside from the above, actual recorded cases of this as evidence leading to a Vestal being accused seem to be lacking. Orosius, in The Seven Books of History against the Pagans, mentions several trials of Vestals for breaking their vow of chastity, but none include a reference to the flame going out. Perhaps this is because it didn't (usually) merit a mention:
Letting the sacred fire out was a serious offence, but the ultimate
crime was to lose her virginity. A non-virgin polluted the sacred
rites and called down the anger of the goddess on the people and city.
Source: T. J. Cadoux, 'Catiline and the Vestal Virgins'. In 'Historia: Zeitschrift für Alte Geschichte Bd. 54, H. 2' (2005)
This is also reflected in the punishment meted out:
Vestals who are guilty of lesser misdemeanours they scourge with rods,
but those who have suffered defilement they deliver up to the most
shameful and the most miserable death.
Other likely occasions when the flame went out would include the sacking of Rome in 390 BC as the Vestals fled the city:
In 390 BCE Rome was sacked by a raiding party from Gaul. While fleeing
the city with his family in a wagon, Lucius Albinius came across the
Vestals, who were likewise abandoning the city with the sacred
objects. He ordered his family and possessions out of the wagon so
that the Vestals would not have to walk with such an important burden.
Source: Roberts (citing Livy 5.39-40.)
Also,
...in 251 BCE, the temple of Vesta was completely destroyed in a large
fire.
Source: Roberts
So, fire consumed by fire? The Temple of the Vestals was also destroyed during the Great Fire of Rome in 64 AD, and again in 191 AD. Presumably, though, the Vestals were not held responsible if the flame did go out in the incidents of 390 BC, 251 BC, 64 AD and 191 AD.
