Hello tspree15,
It'd be a difficult thing to do. Were you looking to do this as some sort of display, or were you planning on using them during a meal?
There are displays where an object (containing an embedded strong permanent magnet) levitates between the poles of a carefully controlled electromagnet. A sensor is constantly monitoring position and adjusting the electromagnet to keep the object rotating. I've seen displays containing a globe, a picture frame, even a gym shoe. While the effect is quite striking, the objects to be levitated need to be very lightweight. I wouldn't think they'd be suitable for supporting a dinner plate with food on it.
On the other hand, take two ring shaped permanent magnets with a (non-magnetic) pole running through the inside diameters. Position the ring magnets in repulsion and attach the dinner plate to the upper ring. With a rare earth magnet one could support quite a bit of weight, but the effect isn't as striking, and the balance may be an issue.
Hope this helps!