What happens to the third hydroxyl group in a phosphodiester linkage?
At physiological pH it is simply deprotonated, i.e. it becomes $\ce{-O-}$ instead of $\ce{-OH}$. That's why DNA is said to have a negative charge.

I realise that this image on Wikipedia omits the negative charge, which is strictly wrong (the negative charge is probably implied because writing it out every time would clutter the diagram). A proper biochemistry textbook will show that it is deprotonated. Here's an image from Berg &amp; Stryer's Biochemistry, 7th ed:



Here's one from Alberts et al. Molecular Biology of the Cell, 6th ed:



Whether you draw the negative charge as being delocalised across the two oxygens (as Berg &amp; Stryer have done) or being localised on one oxygen (as it is in MBotC) is not relevant.