Can a research publication have a bad impact on your career?

First, regarding your comment that “if [the paper] was so bad it would not have been published”: in my experience, pretty much anything is publishable, it all depends on the journal. Don't overestimate the field: there are journals out there that publish absolute crap. And that is not only my opinion, e.g.

in many cases, Bentham Open journals publish articles that no legitimate peer-review journal would accept

Now, the initial question is: how bad can it be? Well, it can be bad. As a researcher (or wannabe researcher), your publications list is like an artist's portfolio. When you look for a job, apply for a grant or try to recruit new team members, this is what they will judge you on.

Now, for a young student (say undergrad or MSc level), one weird paper would not completely freak me out. It would mostly speak to me about the quality of the tutor/advisor/program director: a newbie can sure get a wrong idea at some point, but how come noöne was there to tell him it was a wrong idea? On the other hand, if he was told and did not listen, it might speak to his character!

Not if you published it as a non-peer-reviewed technical report / note on your webpage, or even a adapted it into something like a blog post.

However, it's quite often the case that small results can be augmented with related small results, or added to more significant results, too build up a paper that' more worth of publishing "for real", i.e. in a respectable conference or journal.