Learn Russian: Фильм кончается поздно.

You are right, кончается is in the present tense (ends). To change this to the compound future tense taught in this lesson, you add the conjugated helping verb быть (to be) before the infinitive of the main verb: Фильм будет кончаться поздно (The film will end late).

Why does the verb кончается have -ся at the end?

The -ся ending makes the verb reflexive. We use it here because the film is not ending something else; it is simply ending itself. If you remove -ся, the verb means to finish something, like when a person finishes a task.

When making the future tense version (будет кончаться), why does the infinitive end in -ся instead of -сь?

In Russian, reflexive verbs add -ся after a consonant or a soft sign (like the ь in the infinitive кончаться) and -сь after a vowel. Since the compound future always uses the dictionary infinitive form of the main verb, it keeps the -ся.

Is поздно a neuter adjective here?

It looks exactly like one because of the -о ending, but поздно is actually an adverb meaning 'late'. Adverbs describe actions (like when a film ends) and they never change their shape. It will always stay поздно regardless of what the subject of the sentence is.