I have tile that is on the wall behind my bathroom sink. I would like to remove it and paint the wall. How do I get the old dried grout off the wall?

When you remove the tile there will be significant damage to the drywall. You did not mention if you tile is plastic or ceramic but both will not come off clean. If you are lucky you might be able to flatten the wall with joint compound. Sand the compound, prime and then paint.
If your sink is attached to the wall itself you will need to temporarily remove the sink and the mounting hardware.

I agree although it may be more trouble than its worth, re-drywalling qwould save a lot of scraping and sanding..

Gunsguy

I agree that this type of patch is not always worth the work. A lot depends on how the fixtures are mounted. Redrywalling is quite difficult if you have to remove the toilet and or sink cabinet. I'd personally go for new tile rather than trying to paint.

I find it easy to remove old grout by using a wallpaper steamer on the same spot for a while which softens the grout & is a lot easier to scrape off. Well worth a loan or even purchase of a wallpaper steamer.

I have tried the tip about using a wallpaper steamer on the grout, and have cleaned off a whole wall today. Great tip, thanks.