As someone who has tattoos, piercings such as snake eyes, large gauges, lip piercing, bullring etc I can tell you from personal experience that in nursing school you will likely be subjected to a strict dress code, either by the school or your clinical sites. I tell prospective nursing students to let their personal identities go for the time being, you are learning to become a nurse and should focus on adopting that culture instead of maintaining your own, it is easier, not required but easier.
As far as the workplace it entirely depends, as was posted before. Your ability to actually function on the job has very little to do with the dress code, it is more of a professionalism thing and what the hospital or facility wants to communicate. Having extremes of anything will be career limiting but not career ending. Be prepared to work with a ton of geriatrics for several years, even if it is just in school clinicals. The endless questions can drive you insane.