Shevuos, Chapter 4, Mishnah 8. Like the previous mishnah, this mishnah too presents us with an example of a situation in which the oath of testimony does not apply.
The case in the mishnah:
A person imposes an oath on two potential witnesses: "Mashbia ani aleichem im lo savo'u useidu li she'ish peloni amar li shehu omeid liten li masayim zuz, velo nasan" - "I impose an oath upon you that you come and testify for me that so-and-so told me he was going to give me two hundred zuz, and he did not give it." The mishnah rules that these witnesses are exempt, even if they denied knowing the testimony.
The reason for the exemption:
As we already saw in the previous mishnah, there is no obligation of the oath of testimony except in a monetary claim resembling a claim of a deposit: an object deposited in another's hands, where if they swore falsely about it they are liable. The basis of the obligation is a monetary claim on actual funds, and where there is no such claim on funds, the obligation does not take effect.
In our case, so-and-so's statement that he intends to give two hundred zuz is not a monetary claim, since even though he said he would give it in the future, he is not obligated to fulfill his statement, and the claimant has no claim against him on its basis.
And even when the claimant is poor and was promised charity:
Several of the commentators note that even if the claimant is poor, and so-and-so said he would give him the money as charity, there is still no obligation here, and this for two reasons:
Beis din does not extract money from him: Beis din does not collect the money from him on the strength of the vow. At most they would employ coercive measures against him - excommunication or a ban - until he fulfills his obligation, but they do not actually take the money from him. This, then, is not a full monetary claim.
The possibility of annulling the vow: In the case of a vow, even when a person vowed to give charity to a particular individual, he can always turn to the annulment of vows and undo his vow. Given this, there is no true claim on actual funds here.
In summary: A person's statement that he intends to give two hundred zuz does not create a monetary claim, and even when it involves charity for a poor person, beis din does not actually extract the money, and the one who vowed can have his vow annulled. Therefore the witnesses are not liable for the oath of testimony in such a situation, even if they denied their testimony.