Nostr: Do I need to pay to use a relay?

Not necessarily.

Short answer

Not necessarily. Relay operators set their own access policies; some are open, some require accounts or payment, and others serve a small community. Check a relay's advertised metadata and terms before relying on it. Paying a relay may fund storage or moderation, but it does not purchase control of the whole Nostr network or guarantee that other clients will use that relay.

Sources

NIP-11: Relay metadata, NIP-01: Basic protocol.