Learn Spanish (Latin America): No creo que ese amor sea real.

The phrase no creo que (I don't believe that) expresses doubt or denial, which triggers the subjunctive mood. That's why we use sea. If the sentence were affirmative, you would state your belief as a fact and use the normal indicative mood: Creo que ese amor es real.

Why are we using a form of ser (sea) here instead of estar (esté)?

In Spanish, we use ser to talk about inherent characteristics or the fundamental nature of something. Being "real" is considered a core, defining quality of this love, not a temporary condition or a physical location, so it requires ser.

Why is it ese amor and not eso amor?

Ese is the masculine singular word for "that" when it comes directly before a noun (like amor). Eso is a neutral pronoun used for abstract situations or unknown objects (like saying "I don't believe that!" -> ¡No creo eso!). You never put eso directly in front of a noun.

Can I leave out the word que, since we often drop "that" in English?