I have a friend with eczema, and she could only use these sparingly. (Not at all if she had eaten oranges recently, oranges are one of her triggers) But occasional use for eye makeup removal or following it up with a moisturizer was fine.
I find the act of wiping off the makeup with a cloth to be irritating in itself. Maybe try a cleaning oil instead? Neutrogena makes one. You just put it on your dry face that has makeup on. The rinse it off and follow with cleanser.
I don't think these are best for sensitive skin. The only ones I found that don't irritate my skin are the Purity ones from Philosophy-although they are more expensive it reflects positively on your skin!!
I have never had an issue with them causing irritation. I am allergic to several different products, but thankfully not these. I will say they will cut though any long wear and waterproof make up! These are my go to makeup remover wipes every night! Hopefully you found this helpful!