You are at the end of your fiscal year, and it is important to your donors that your books are in balance (i.e. no deficit) for the year in...

This is the sort of ethical dilemma that arises in business all the time. This case is interesting because you're not a for-profit corporation simply trying to make money, but a charitable nonprofit that presumably has some sort of social mission.

If you clear the invoice, you will ensure that your nonprofit appears solvent, and the increased donations next year may even make your nonprofit solvent. Since you could pass the audit and have no official reason to know that the invoice will not be paid, you may not even be committing fraud under the law.

And yet, morally, you are clearly lying. You are defrauding your donors, in at least an ethical sense, if not a legal one. You know that the invoice won't be paid, and so by telling them that it will, you are being deceptive.

It could still be justifiable to clear the invoice; many utilitarian ethicists would argue that because your nonprofit does good for the world, you won't get caught, and you will eventually make up the financial deficit, it is justified for you to commit this small act of fraud for the greater good of ensuring the ongoing effectiveness of your nonprofit. That is why it is important that you are not a for-profit corporation; if you were, you clearly ought to honestly take the losses and pass them to your shareholders, as that's how capitalism is supposed to work. But as a nonprofit, the consequences of your actions will ultimately fall upon the people you're helping in your social mission, and it does seem unfair that they should suffer because of this bounced invoice.

Still, most deontologists and even nuanced utilitarians (full disclosure: I consider myself the latter, a nuanced utilitarian) would argue that this act of fraud is wrong, that it is against your fiduciary responsibility as the manager of the nonprofit. Your job is to report the finances of the organization honestly to the donors, and by clearing an invoice you know will not be paid, you are shirking that responsibility. The long-run consequences of a system where people feel free to fudge finances as long as they perceive it to be for a greater good would be disastrous; they would essentially undermine the integrity of our entire financial system. Your heart may be in the right place, but if you clear that check, you're pushing us one step closer to a world of Enrons.