This is a very subjective answer. But I'll try.
It depends entirely on what the tagline is about.
Does it describe the work they do?
Like for example- ABC, interactive production studio.
Does it say a very unique story about the brand in an original way-
How are they using their tagline in the rest of their branding material? Do you see a pattern or thought behind it that they are trying to enforce in the web design? Sometimes they might feel that their tag-line isn't coming across in a powerful way anywhere on the site. For example some agencies show their tagline as a big headline on their about page.
They are probably suggesting the header because they feel that it will be most visible there. But your job is to not find a reason to not put it there. Your job as a designer is to understand why they are asking you to this and then find a way to solve that problem rather than arguing why the header is not a good place. Maybe it could be a headline on your front-page or somewhere in the footer. Many possibilities. But they all begin by understanding your client's relationship with the tagline and how to best put it across. Hope this helped.
Some more examples- example 3, example 4