Learn Spanish (Latin America): Hace falta más tiempo.

Both express that something is required, but hacer falta is a highly conversational and natural way to express that something is missing, lacking, or necessary. It translates roughly to "to make a lack."

What is the grammatical subject of this sentence?

The subject is actually más tiempo. The phrase hacer falta works backwards for English speakers, exactly like the verb gustar. The thing that is lacking or needed is the subject performing the action, which is why the verb is the singular hace.

If I wanted to say "More hours are needed," how would this sentence change?

Because the thing needed is the subject, the verb must agree with it. If you change the singular más tiempo to the plural más horas, the verb must also become plural: Hacen falta más horas.

How would I change this sentence to say "I need more time"?

Since it functions like gustar, you just add an indirect object pronoun before the verb to specify who is experiencing the need. You would say Me hace falta más tiempo.