A regular long sword would almost certainly be considered a metal weapon, but a Sun Blade is just the hilt with a magical blade. Longsword hilts were made of wood, wire, and leather, according to this HowStuffWorks article about sword-making.
The description of the weapon is mum on the materials of the hilt so I think the DM needs to make a call on whether or not the heat metal spell has any effect.
One point to remember here: we're dealing with a game of magical fantasy. So while real world proxies can help inform the imagination, they shouldn't limit its scope.
I would argue that the hilt is made of metal, at least partly. This is based solely on the fact that almost all swords have a metal component to the hilt or tang, even if the hilt or tang is encased in leather or a decorative material. In other words, holding this thing would get very hot if the spell heat metal was cast upon it.