Nostr: Do I need a Lightning wallet to use Nostr?

No.

Short answer

No. The basic protocol for identities, posts, follows, and relays does not require Bitcoin or a wallet. A Lightning wallet is only needed when you want to send or receive Lightning payments such as zaps. Choose a wallet and connection method deliberately, review spending permissions, and do not treat a Nostr app as a wallet unless it actually provides wallet functions.

Sources

NIP-01: Basic protocol, NIP-57: Lightning zaps, NIP-47: Nostr Wallet Connect.