In January of this year, a new website of one of our clients went live. They had an different website before that. So I will refer to the old one, as the previous website, and the new one as the one that went live in January.
The problem now is that some websites still see the old website. For example, we have a RSS feed for Hootsuite. But Hootsuite doesn't see the RSS feed as a valid one. So I checked the RSS feed using http://feedvalidator.org/, and came to the conclusion that it renders the old website. That results in a 404 error, because they didn't have a RSS feed on the old website. If you view the feed page in the browser, it works fine.
I have no clue as to why it still renders the old website sometimes, and I don't know where (and how) I should look for a solution. So I hope someone can help me, or at least point me in the right direction.
After endless tries with no result, I found the Facebook Developer Community on Facebook. When I posted my question there, someone answered:
"I just checked your DNS and everything seems to be fine except for 1 thing. The domain name has an AAAA record (IPv6) which points to 2001:888:0:18::117:80. After checking, your server does not seem to listen to this IP6 address (domain AAAA not/misconfigured?). Facebook always tries to use IPv6 before IPv4, so that might be an issue there."
We don't use IPv6 yet, so I knew there was something wrong with the IPv6 part. Turns out their old hosting did have IPv6 records. When I removed them, the site worked fine and Facebook finally saw our new site!
So, for some reason, some providers still had very old DNS records cached (for the old hosting company) instead of our new DNS records.
I answered this on my own question, in the hopes that it might help someone in the future who stumbles upon this question.