Symptoms

DNS forwarding does not work for a website and the following error appears:

404 Server Not Found

DNS name is not resolved to Plesk IP address by public DNS servers:

# ping example.com
PING example.com (203.0.113.2) 56(84) bytes of data.
64 bytes from external.tld (203.0.113.2): icmp_seq=1 ttl=54 time=233 ms
64 bytes from external.tld (203.0.113.2): icmp_seq=2 ttl=54 time=233 ms
64 bytes from external.tld (203.0.113.2): icmp_seq=3 ttl=54 time=233 ms

DNS name is resolved to Plesk IP address by public Plesk DNS server:

# dig @203.0.113.5 +short example.com
203.0.113.5

Cause

Domain name is resolved to incorrect IP address by public DNS servers. Misconfiguration on domain registrar side.

Resolution

Contact domain registrar company where the domain name was purchased to set the correct IP address on the DNS zone. Name server record ns.example.com should be associated with the IP address that was set in Plesk > Domains > example.com > Websites & Settings.

After the configuration changes are made on the domain registrar's side, it takes up to 48 hours for them to get propagated (applied).
As a temporary workaround for the clients, the correct "IP address - domain name" association can be added to %systemroot%\system32\drivers\etc\hosts file on their workstations:

203.0.113.5 example.com

Note: the actual domain name and IP addresses should be used when applying the instructions.