308 preserves the HTTP verb[1], and the form on that page uses POST. POST is not idempotent[2], which means using it more than once with the same parameters may not yield the same output. For example, POSTing this comment form twice would append to the resource twice; as opposed to GETing twice which just returns the resource unmodified both times.
Firefox correctly (under the old RFC for a 301 redirect[3]) asks for confirmation before automatically repeating a request that is not guaranteed to be safe to repeat. Some implementations will instead convert the request into a GET, which is why 308 was needed in the first place.
RFC 7238 with the 308 status code isn't part of HTTP/1.1 (even under the new revision, which is in RFCs 7230-7235), its an experimental extension that expressly notes that implementers must be aware that HTTP/1.1 clients not specifically written to that extension will fall back to the behavior for status code 300 when status code 308 is encountered, and that 308 should not be used where that behavior is not acceptable. (This is the standard mechanism for extensibility in response codes within the existing high-level groupings in HTTP/1.1.)
I get failures in Chrome and IE. Firefox passes if I click OK in a funny dialog.