Question
Do you have to cut down a pine tree when there is woodpecker damage?

Answer
If the needles are green then the tree is ok. What you are describing is made by a woodpecker (bird) called a Sap sucker. These birds make the holes and insects will come to feed on the sap and the bird returns to eat the insects. The white is sap from the pine You can spray the bark where the holes are located with an insecticide and this will kill the insects and the birds will stop coming. Use any house hold insecticide like Raid or Sevin.
I would also fertilize the tree with 10-10-10 fertilizer at the rate of 1 lb of fertilizer per inch of trunk diameter scattered around the tree and watered in good. Do this now and again next spring.