Unfortunately, we have no example of this ever happening. Plutarch (Life of Lykourgos 6) is the main source for the institutions of Spartan government, but he does not resolve this question. He does include some speculation as to why the gerousia consisted of 28 men in addition to the 2 kings:

The number of the elders was fixed at 28 because, according to Aristotle, two of the 30 original associates of Lykourgos abandoned the enterprise from lack of courage. But Sphairos says that this was originally the number of those who shared the confidence of Lykourgos. Possibly there is some virtue in this number being made up of 7 x 4, apart from the fact that, being equal to the sum of its own factors, it is the next perfect number after 6. But in my own opinion, Lykourgos made the elders of just that number in order that the total might be 30 when the two kings were added to the 28.

-- Plut. Lyk. 5.7-8

Interestingly, while there is much guessing here as to the administrative or mathematical principle behind the number, there is never any question as to why the number is even. This despite the fact that the Athenians, apparently more aware of the problem of a split chamber, always made sure their juries were of an uneven number (typically 201 or 501).

The probable reason why Plutarch doesn't question the number or explore the possibility of a tie is that in political matters, the gerousia acted as an oligarchic council, which was expected to come to a consensus before putting its proposal to the Assembly. As a small group composed from the very highest echelons of Spartan society, it is likely that they would usually be able to reach a common accord. Plutarch (Lyk. 5.7) argues that they tended to form a bloc to protect the kings against the Assembly, and the Assembly against the kings.

In legal matters, it seems the number of the gerousia was a non-issue, because the highest court of Sparta consisted of the elders and the ephors, who were 5 in number. A late source (Pausanias 3.5.2) reports a case where one of the kings was put on trial, and the gerousia was in fact split down the middle in its verdict, with 14 elders and the other king voting that he was guilty. However, the ephors tipped the scales decisively in favour of acquittal.

Considering the obvious advantage of an odd total of votes, and considering the function creep of the ephorate after its introduction at some point in the 6th century BC, it is plausible to assume that in case of a tie in the gerousia, the ephors would be called in to cast additional votes. But this point is pure speculation; we do not know if this ever actually happened. Spartan politics were renowned for their stability and apparent harmony, which in itself may have been enough to dissuade voices of dissent in the council of elders.