You will have a mix of young and old professors - no way to generalize. I've always hidden mine out of general paranoia / practicing for work, but I don't think there's a great reason. If you're planning to seek a recommendation from a professor at some point, I would maybe cover them up to be extra careful. So maybe if you want to clerk or transfer.
Employers don't care that you HAVE tattoos, but it's also more than fear of a client seeing them; in most firms, they should never show at the office, even if there are no outsiders there. And I would avoid letting anyone from work know about the tattoos at all, at least for a few years. I guess law school draws from some populations more than others, but I have been surprised by how conservative some of my classmates are. Not like politically, but like 'omg tattoos are a big deal' conservative. Different worlds I guess. So if those are the kinds of people you're going to be working with, you should probably keep tattoos under wraps, literally and figuratively, until they already know you're smart and serious and competent. And again, even after people know, they should never show at most offices.
So you'd advocate being on the "safe side" eh? And the students you've met are shocked by tattoos? What school did you go to/are you attending?
BTW, is your AV the strange room from David Lynch movies?
Not shocked, or mean really, just a little more judgmental/surprised than I'm used to. I go to NYU. I'm just used to the idea that in fact everybody and their mother has a little tattoo somewhere, but some people here are like 'gosh no, I don't know anyone with tattoos,' or 'but you'll be a lawyer!' - like there isn't such a thing as long sleeves. Most people don't know that I have tattoos though, so that's a really small sample size. And I do know a couple people who also have tattoos. But yeah, I'm definitely a safe-side person when it comes to career stuff. ITE and in a conservative career, why risk it.
And yes, it IS the strange room! I think of it as the waiting room. Anybody who recognizes a David Lynch tar is awesome as far as I'm concerned.