This question comes up a lot.
I generally recommend that people keep them covered for interviews and clinical activities, just because you don’t know what your interviewers’ or your patients’ preferences and sensitivities would be. But tattoos really shouldn’t affect anything. I know LOTS of docs, nurses, pharmacists, and other hospital staff with plenty of tattoos (even full sleeves). Our hospital requires employees to keep them covered at work, but they don’t mind you having them, and it doesn’t affect employment.
I work with one male physician who has pierced ears and pierced nipples that are generally pretty visible under his ratty Affliction t-shirts. On weekends he walks the halls with ripped jeans, boots, and a leather jacket, and patients who don’t know him assume he’s a visitor, but he’s an excellent physician, so no one says anything about his wardrobe choices. Ultimately, what matters is whether you’re good at your job or not. Ok, but maybe don’t go this far.