Toolset Forms includes post expiration functionality.
You need to enable it in the settings at Toolset > Settings > Forms and then on any form you can specify a period after which posts will expire.
For this to work the posts need to be published by the form, but can be edited in the backend subsequently (including changing the date when the post expires).
Nigel is away and will be back tomorrow, but I'll be happy to follow up.
From your screenshot, I'm afraid, it is not clear where and when the expired link message is showing.
The "Toolset Forms" is available as a separate plugin and to check if it is installed and activated on your website, you can go to WP Admin -> Plugins -> Installed Plugins.
( screenshot: hidden link )
To programmatically expire all existing and new posts, you'll need custom code. For more personalized assistance around custom programming, you can consider hiring a professional from our list of recommended contractors: https://toolset.com/contractors/