I'm currently interning for a small R&D company. The boss is notoriously tight fisted and prone to cutting corners to save any amount of money he can. In the past, he has attempted to use my student status to get free or discounted software for company use to avoid paying licensing fees. (To clarify, software purchased using my status as a student but not to be used by me but by other individuals in the company) He recently found out I get institutionally provided access to various journals and has started requesting I provide him with pdf or printed copies of papers so he can avoid paying for them. Is this ethical? Legal?
It is unlikely that your use of this journal subscription would fall within the terms-of-usage agreed by the university with the journal providers, and agreed by you (as a student) with the university. You should enquire further into the particular terms of access for these resources, but I would imagine that they would be restricted to use for educational and research purposes within the scope of your position as a student at the university.
Either of these practices could potentially get you into trouble, though the use of software beyond the terms of the licence is probably more fraught with danger. Downloading journal articles for work purposes is not that uncommon in fields employing academics and students, though it is probably not legal. While some forbearance might be given for small infractions, certainly it is unethical to agree to access a resource from the university for the purposes of your education, and then to use that resource systematically to advantage a commercial business that is unwilling to pay for the service.
As to downloading articles, it's not ethical in one sense. However, many research articles are funded by public dollars. Is it ethical that they wind up behind paywalls? Institutional access to journals costs a lot of money.
Yes, the publishers are providing a service by disseminating the material, and yes, that deserves remuneration. However, once again - most research has some public funding behind it.
In my view, depending on the context, asking a student intern to download a few articles is not necessarily unethical. If unethical, it's not necessarily the sort of thing that you want to call the cops for. Asking an intern to buy student software for the rest of the company is definitely not ethical. That plus the OP's tone makes me think there is more stuff going on that he or she is not saying. In ethics, your gut does have to play some role in making your decision.
Why are you asking whether this is unethical? That's something for you to decide. Possible points to consider: is the paywall system underlying most academic journals ethical in and of itself? Is being "unethical" towards something which is unethical actually something that is really unethical?
Is it legal? No.
Based on the title of this question, of course not. It's violating terms of service not just unethical.