Thanks for the additional details. Let's dig in and figure out what's going on. :)
What is strange is that once I click on customize, access to the dashboard disappears. That's not normal is it?
Yes, that's actually normal. I think I missed your comment about this earlier — sorry about that! The customizer will give you a live preview of the changes you're making, along with a side panel with all of the customizer controls. We have some additional information about that here: Customizer
I go to "colours" while in customize and press "save" - essentially do exactly what the tool says - and then I refresh the page (since I can't go back to the dashboard) Changes disappear to show a change from yesterday morning i.e. white text on the header and beige background. This template keeps coming up whatever I do.
When you're in the customizer, you can click "save" to save your changes and then click "cancel" to leave the customizer tool. That will take you back to your dashboard. Clicking refresh while you're in the customizer may make things a bit wonky. :)
If that doesn't make it work for you, can you please let me know exactly what change you're making (for example, what color you're selecting or what header text setting you're choosing)? I understand where you are making the changes, but the exact setting you're changing and what you're trying to change it to can make a difference. That will also help me test the exact change you're trying to make to see what's going on.
I have the Suits theme up on my test blog. I am running Firefox 26.1 on Windows 7 and I can change the header font color at > Appearance > Header
__Show header text with your image.
However, that change is not reflected on the blog so if the theme has this feature then there is a bug.
Timethief, did you select any custom colors (e.g. the featured color pallettes) in the customizer? If so, those colors will override any header font color selection under Appearance > Header.
I tested with the Suits theme default colors, and changing the header font color under Appearance > Header worked for me. I saw the change reflected on the blog. I'm happy to keep testing, though!