Unfortunately, there’s no direct way to delete threads with Disqus.
However, the following is the best workaround possible in order to pseudo-delete them:
If it’s important to you that the comments themselves are deleted and not be accessible anywhere, you’ll need to delete them individually first, from: https://yoursite.disqus.com/admin/moderate/#/all.
Replace yoursite with your site slug.
Create a new page on your website: https://yoursite.com/removed/, and block it from being indexed in search engines with: <meta name="robots" content="noindex" /> (because this page will serve no purpose to being seen by your site visitors).
Go through all threads on https://yoursite.disqus.com/admin/discussions/, one by one, page by page, and wherever you find a thread for a page that has been deleted or that you otherwise wish to remove, change the URL to: https://yoursite.com/removed-1/, changing the number for each one, https://yoursite.com/removed-2/, https://yoursite.com/removed-3/, etc., for as many threads as there are that you wish to remove (you might be surprised at how many are floating around).
Open up Notepad or TextEdit and add the following:
https://yoursite.com/removed-1/, https://yoursite.com/removed/ https://yoursite.com/removed-2/, https://yoursite.com/removed/ https://yoursite.com/removed-3/, https://yoursite.com/removed/
I’d recommend preparing your first line with the correct site info, then dupe that 9 more times so you have a set of 10, add a blank line at the bottom, copy that, and paste that over and over for as many threads as there are that need to be removed. Then go through each line, one by one, numbering them accordingly, remove the blank line at the bottom when done, and save as a text file, delete.txt or whatever is fine.
Now, go to: https://yoursite.disqus.com/admin/discussions/migrate/, click “Start URL mapper”, and submit your text file.
That’s it, you’re done. Note that it’ll probably take a few hours to complete. Technically, you didn’t really delete the threads. More so, you merged them all into one thread. You might also want to make sure to close that thread. Anything you want to delete in the future, just merge into https://yoursite.com/removed/.
Not the prettiest solution, but probably the best tutorial written to date on the only option available to us.