A van weighs 10,000 pounds and has 20 pounds of birds perched in it. The combined weight is thus 10,020 lbs. The van is sealed airtight.

If the birds all take off and are flying around inside the van, then how much does it all weigh?

The same - 10,020 pounds. Flyng around doesn't make the birds weigh less.

To clarify the question: if the van is sitting on a scale while the birds are flying around inside, what would the scale read?

Most people are surprised to learn that the answer is that the scale reads the same regardless whether the birds are sitting or flying - in this case 10,020 pounds. The reason is that the force that keeps the birds aloft must cause an equal and opposite reaction (good old Newton at work), which means that the air under the birds' wings is compressed and presents a downward force on the floor of the van that is equal to the bird's weight. The result is that the scale measures the sum of the van's weight plus the downward force caused by the birds.

As a corollary question - if the birds were strong enough and pressed up against the roof of the van could they get it to lift off the ground?

Nice explanation EB!

Thank you... I win the bet!