DIAGNOSIS That's heat distortion, not melting—yet. The side tables are made of a heat-resistant plastic, but prolonged, direct heat from the grill body or sun can cause them to soften and sag.
FIX First, always fold the tables *down* when the grill is in use or stored in direct sun. The folded-up position exposes the underside to radiant heat from the grill body. If it's already warped, you can sometimes gently reshape it. With the table removed (it pops off its hinges), lay it flat on a counter with the warped side up. Place a flat, heavy object (like a stack of books) on the high spots and leave it for a day or two in a cool room. It might not become perfect again, but it can improve.
IF THAT DIDN'T WORK The tables are designed to be removable and replaceable. You can order new ones directly from Weber's parts website. They're a straightforward snap-on installation.
HEADS UP Never place hot pots, pans, or freshly taken-off grill lids on the plastic tables. Use the built-in lid holder on the side of the grill for the lid, and a separate surface for hot cookware. The plastic just isn't rated for that kind of direct, concentrated heat.