Yes. It is normal to feel burning when you get shampoo in your eyes because it will affect the ph balance of your fluid in your eye. However, after you rinse them with water, your eyes will get healed soon with no damages.
If the PH level is lower or higher than standard level, your eyes will feel burning. You should choose shampoo which is mild chemically. If you wash them out immediately with fresh water, they should not damage your eyes.
Most the shampoo will cause no damage to your eyes if it gets into your eyes. On the bottle you will find the suggestions saying that you can rinse your eyes with clean water if you get shampoo in eyes. However, If it is serious, you should go to see your doctor or optometrist immediately.