I have an apt with my own entrance in a private two family house.

In the bathroom there is a constant (never stops) sound of something
Like running water... I hear it behind the toilet. My landlord says
He has the same sound but won't call a plumber because he sees no
Leaks.
Can he be missing an important defect?

To Mark: You keep mentioning tank but I don't have a tank. There's a pipe from the toilet to the wall.
I don't have the type of toilet with a tank. I didn't want to click not helpful because you certainly
Took the time to explain the problem and how to determine if this is a problem but...

My landlord finally woke up. The plumber is coming tomorrow

Perhaps Mark would have been more helpful if you would have furnished him with all the information up front.
We're good but not good enough to to answer question flying blind without all the facts. You're welcome to post questions but next time please give us all the details in your first post.
Had you clicked on not helpful and given Mark a reddy for your oversight this post would have been much different.
I hope this helps, Tom

Tom: I didn't click yes or no... for that very reason. I didn't know that most people have tanks

No problem Jammy.
If you have more plumbing problems please give us all the details. The more we know about your problem the better eequipped we are to suggest a solution. If you have more problems we're as close as a click. Cheers, Tom