Turning website chat off
Remove the chat from your website and free its channel slot; your chats and code are kept.
Updated
On the settings page, press Turn off. You're asked "Turn website chat off?": the chat button disappears from your website and the channel slot is freed. Changes reach your website within a minute. Chats stay in the Inbox, and turning it back on keeps the same code. Press Turn off again to confirm, or Keep it on.
The code can stay on your website: it shows nothing while website chat is off. To use it again, open Channels → Website chat and press Turn website chat back on, which needs a free channel slot. Your settings and the code are kept.
Without a plan the chat is hidden the same way. When a free trial or a plan ends, the button disappears from your website until the owner chooses a plan, then comes back by itself, with each visitor's chat still there.