No, it will not harm your straightening but it's a good idea to always rinse the chlorine out of your hair as soon as you can to help keep your hair from drying out. I wouldn't suggest going in the chlorinated water for 24 (or to be more safe) to 48 hours after any chemical application whether hair straightening, curly perm or a color though.
By
05/22/2011
I have a professional beauty shop perm in my hair and pool chlorine has never ruined my perm. It shouldn't have any effect on your straightened hair.