Lightning network wallets are "hot" - the nodes are always online by design. This is what allows routing payments across nodes to occur.

If your wallet uses a private channel with a LSP (lightning service provider), you might just have an app on your phone, but you still have funds locked in a time locked contract with your peer. At any point they can try to close the channel, and could theoretically try to cheat you by broadcasting an old channel state (say the funds were all on their side of the channel at the beginning, but you've gradually accumulated funds on your side over time - they could try to broadcast the original state instead of the current balances.) Because of this, your wallet needs to be online regularly to make sure this doesn't occur. (Or use a watchtower to check for attempts to cheat you on your behalf.)

If you're using a custodial lightning wallet, it can manage channels without you, but it's obviously not a trustless way to store funds long-term - the custodian could simply shut down and run off with your funds at any time.

If you keep your node up to date and connected, you're generally okay, but there have been exploitable vulnerabilities in the past, so being able to patch your node software quickly is recommended, and operating one is not without risk.

Contrast this with offline cold storage - a hardware wallet or simply an offline seed phrase / codex32, and there is much less to go wrong for long term storage.