No, you don't HAVE to use it. But at least one coat is recommended even under a moisture resistant paint. To protect the shelves of my rack, which are plywood, I used plastic sheeting on the bottom of each as well as a single coat of poly, then painted the remaining exposed wood with Rust-Oleum latex based paint. So far so good.
And, between MDF and Melamine I would go with melamine. It's sturdier and slightly more resistant to moisture, tho just as heavy.