Great question. The answer depends entirely on the plugin.
Generally speaking, you should be able to update a theme or plugin without causing a disruption on your website. The developer of a well-coded plugin will have taken updates into consideration, and carefully tested it before releasing a new version, to minimize any bugs or issues related to updating.
Your best bet is always to create a backup of your site first, then update your plugin(s). That way, if you encounter an issue, you can easily revert back to a prior version, and then notify the creator of the plugin for support.
Hope this helps!