I imagine you've already investigated, but are there any firmware or software updates on the Dell Zino website to get this working? I can't imagine they'd ship a product with hardware that isn't functionally available in the OS
Sounds like a faulty mobo. I have been reading up AVS forums who now have a 90 page thread on Zino. Everyone who got their's delivered had HDMI work out of the box.
All had problems with over scan with HDMI connection (black bar's on side/bottom/top) of TV and they had to install the latest catalyst drivers, some had to install the latest video drivers from Dell's website.
I believe Dell's standard 1 year warrnaty is in-home service if they fail to resolve the issue via their remote desktop software.