CounterSliver should do fine against Burn, since it has a decent clock backed up with countermagic. That's how Gro beats the deck.

Try sideboarding out Crystalline Slivers for Meddling Mages. Making your guys untargetable is poor because it shuts off your ability to Swords them for life, and because you should almost always be happy if a Burn player aims a burn spell at one of your creatures instead of at your face.

Mage is nice because it can name Flamebreak, which should often force the opponent to spend a card removing it.

Burn is a combo deck without any draw, which means that it loses easily if you force it to interact. It can't afford to actually trade cards with a deck with superior card manipulation and more effective damage sources (creatures).

If you really want to board for the deck, though, Blue Elemental Blast or Chill would be the way to go, depending on which you find is better against Goblins (since that's a real deck you should actually care about).