Spawn Tunnels only appear in Bug Heroes, on a somewhat rare quest. The tunnel is a hole in the ground and looks extremely similar to Ant Hole, one of Ant's abilities. It will constantly spawn one different type of enemy each; most likely Grubs or Fleas.
They only appear when the quest "Enemy Reinforcements" is accepted. To complete the quest, you must destroy five of these Spawn Tunnels. The battle can get very hasty and messy, with grubs constantly crawling over to the base. A great stragety is to have someone with a turret ability, particularly Termite. If turrets were placed all around the map, they might be able to destroy the Spawn Tunnels for you. Otherwise, you could very easily lose the game, especially if you accepted during the middle of a hard round.