Can I run a program made in Delphi 6 in newer versions of Delphi?

In theory, yes, but it may depend on the specific code. There are some little things that have become incompatible, so just trying to know. Even if something does not work, it's quite simple to adapt to work.

Obviously you will not have access to new features automatically. Some maintenance can become more complicated. If you are using third-party external components it can be a lot more complicated.