Learn Spanish (Spain): Eso es falso.

Ese (masculine) and esa (feminine) are used when you are referring to a specific noun, like ese libro (that book) or esa gata (that cat). Eso is a neuter word. It is used when you are talking about an abstract idea, a general situation, or a statement someone just made. Since "that" here refers to a general statement rather than a specific object, we use eso.

Since we are learning about negation, how would I change this to say "That is not false"?

To make a sentence negative in Spanish, you simply place the word no directly in front of the verb. The verb in this sentence is es (is), so you would say: Eso no es falso.

Why do we use es (from the verb ser) instead of está (from estar)?

A common rule of thumb is that ser is for permanent things and estar is for temporary states. While you might think a statement's truth could be debated, in Spanish, being true or false is considered a core, permanent characteristic of that specific statement. Therefore, we use es.

If I wanted to use negation to say "That is not true", how would I say it?

You would still put no before the verb. The most natural ways to say this in Spanish are Eso no es verdad (literally "That is not truth") or Eso no es cierto (That is not true/certain). Both follow the exact same negation rule.