If the pool water is properly maintained, they shouldn’t.
We add chemicals to swimming pool water to control microbes like bacteria and algae. Chlorine can irritate eyes, but that is usually an indication that too much was used. A chemical called cyanuric acid is added to pool water to protect pool chlorine from sunlight. Cyanuric acid forms a weak bond with free chlorine in the pool water, preventing it from evaporating out of the pool. It also keeps free chlorine from irritating your eyes.
If there is too little chlorine in the pool water, this can cause cloudiness and the production of chloramines (that is what smells like chlorine to your nose). These can cause eye irritation.
Other things that can irritate the eyes are pH imbalances and hard water. We add acid or alkali to swimming pool water to adjust the pH to a neutral level. If this is done properly and often enough, eye irritation due to acidic or alkaline water will not be a problem.