f a computer (running different services such as ftp and ssh) is
placed on a Demilitarized Zone, will all of its services automatically
be accessed by anyone on the internet?
That depends upon the rulebase already in place on the firewall. However, the answer to your question is most likely no - based on the assumption that a NAT rule will not be in place for a new asset.
If the answer is yes, then is it possible to only allow a specific
service such as ftp, to be accessed over the internet and block access
to all the other servers?
Yes, this is possible. Ideally there would also be switch ACL's in place that would prevent the assets in the same DMZ from communicating with each other as well.