The birds beat their wings against the air. The air presses against the walls of the truck.
*If* the walls are continuous, with no gaps, the air pressure in the truck would rise. But the truck would weigh the same, because the birds are supported by the air, which is supported by the truck.
If there are air holes in the walls, then the air can escape in high presure regions. More air will be drawn in at low presure regions. The birds are supported by the air, which in turn is *not* fully supported by the truck. The weight of the truck *will* drop. The drop in weight will be large but will not equal the full weight of the birds however, because the air being forced out of the truck will still exert a small force on the truck.
There ya go