Learn Italian: È un gioco noioso, ce ne si stanca subito. - It is a boring game, people get tired of it immediately.

The verb here is stancarsi (to get tired of). To make a reflexive verb impersonal ('one gets tired'), Italian changes the impersonal si to ci to avoid saying si si, resulting in ci si stanca. When you add ne ('of it'), the ci becomes ce, and ne is placed before the reflexive si, creating the cluster ce ne si.

What exactly does ne replace in this sentence?

Here, ne replaces di questo gioco (of this game). The verb stancarsi requires the preposition di (as in stancarsi di qualcosa). Because ne is used to replace phrases starting with di, it steps in to mean 'of it'.

Could I say se ne stanca instead of ce ne si stanca?

Yes, but it changes the meaning. If you are talking about a specific person, you don't need the impersonal pronoun. The reflexive pronoun si changes to se before ne, giving you se ne stanca (he or she gets tired of it). The ce in this sentence is only there to add the impersonal 'people' or 'one' meaning.

How would this look in the past tense ('people got tired of it')?