Short answer: No, WordPress doesn’t care about pastebin in particular and doesn’t do any sort of filtering like you’re describing.
Long answer: Your host is doing it.
Step 1: Ask your host if they have mod_security enabled and what their filtering rules are.
Step 2: Try pasting the same link via a comment or something. If it’s being blocked on any POST attempt (which is what it looks like to me), then it’s due to your server doing the filtering, not WordPress.
Also, next time, instead of posting a three and a half minute long video with useless music instead of a voiceover describing the problem, perhaps just saying something like “anytime I post a link to pastebin that looks like a working link, my site shows me a 404 error on /wp-admin/post.php” would be a bit more useful and more likely to receive answers. It’s kinda spammy behavior to post videos like this.
Just a tip for the future.