I'm not sure what you mean by the "remote channel". If you don't control the channel and you grant full access to a third party to manage your Lightning node/channel then the worst they can do is spend all the Bitcoin in that channel and push the total capacity to the counterparty's balance in the channel. You can effectively lose all the Bitcoin you initially locked up in the channel.
Assuming you control the channel, if funds are moving back and forth in the channel the Lightning security model relies on observing the blockchain in case your counterparty tries to cheat you by posting a revoked state. If you don't do this or stay offline for a sustained period you may miss the cheat attempt and hence fail to respond within the required time window by broadcasting the justice transaction. The amount of Bitcoin you lose will be the difference between your current balance in the channel and the balance you had in the revoked state.
If you are a routing node and not just an edge node then being offline for a sustained period could also result in you not earning routing fees you could have earned in that time. Other Lightning nodes may choose to close channels with you as you have potentially cost them routing fees in the time that you've been offline. Rather than revealing the preimage to the hash of the HTLC in a timely fashion you may have locked up capital across the route, prevented Lightning nodes from routing other payments and ultimately require them to close channels. As a routing node you should really aim for 100 percent uptime. As an edge node you are just hurting yourself if you are offline for a sustained period as you can't receive payments and you are giving your counterparty the opportunity to cheat you.