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Appeal to spite (also called argumentum ad odium) is a logical fallacy in which someone attempts to win favor for an argument by exploiting existing feelings of bitterness, spite, or schadenfreude in the opposing party.

Description

The Appeal to Spite Fallacy is a fallacy in which spite is substituted for evidence when an "argument" is made against a claim. This line of "reasoning" has the following form:
    Claim X is presented with the intent of generating spite.
    Therefore claim C is false (or true)

This sort of "reasoning" is fallacious because a feeling of spite does not count as evidence for or against a claim. This is quite clear in the following case: "Bill claims that the earth revolves around the sun. But remember that dirty trick he pulled on you last week. Now, doesn't my claim that the sun revolves around the earth make sense to you?"

Of course, there are cases in which a claim that evokes a feeling of spite or malice can serve as legitimate evidence. However, it should be noted that the actual feelings of malice or spite are not evidence. The following is an example of such a situation:

Jill: "I think I'll vote for Jane to be treasurer of NOW."
Vicki: "Remember the time that your purse vanished at a meeting last year?"
Jill: "Yes."
Vicki: "Well, I just found out that she stole your purse and stole some other stuff from people."
Jill: "I'm not voting for her!"

In this case, Jill has a good reason not to vote for Jane. Since a treasurer should be honest, a known thief would be a bad choice. As long as Jill concludes that she should vote against Jane because she is a thief and not just out of spite, her reasoning would not be fallacious.
 

Examples

Bill: "I think that Jane did a great job this year. I'm going to nominate her for the award."
Dave: "Have you forgotten last year? Remember that she didn't nominate you last year."
Bill: "You're right. I'm not going to nominate her."

Jill: "I think Jane's idea is a really good one and will really save a lot of money for the department."
Bill: "Maybe. Remember how she showed that your paper had a fatal flaw when you read it at the convention last year..."
Jill: "I had just about forgotten about that! I think I'll go with your idea instead."