What are some examples of negative effects on a career for boycotting Elsevier journals?
People decline to serve as editors or reviewers for all sorts of reasons. "Too busy" and "conflict of interest" are two common ones that are easily applicable to your case.

All junior faculty have too much on their plate and no one knows what your chair or mentor has told you that you need to focus on. So "unavailable/too busy to take this on" is an unprovable and thus good excuse. It's also impossible to know who you've had a fling or a spat with and so "conflict of interest" (and you don't have to give the reason for a CoI or for why you're unavailable) is perhaps less apt in this situation but also a valid excuse. 


  Example: These are the three options I was given when I received my last reviewer assignment email. If I clicked on decline due to COI or unavailable, I'd be taken
  to a webform that asked me for alternate reviewer names, but that was
  entirely optional on my part. I would not be asked for a reason or justification for my response.
  
  
  


As to negative effects, I think it's fine to have principles, but when you're junior faculty, you're also vulnerable. You can choose not to buy products made in China (for example) or to decline to provide free work to Elsevier, but you also don't have to tell any and everyone that you've made that decision for yourself. It's fine to decline with a vague reason if that will protect you from the  possibility of retribution.

When you are in a position where you can speak your mind without fear of retribution(which is what tenure is designed to ensure), then do so.

As an aside: Especially in well-established journals, editors and editorial boards tend be composed of people known for their expertise as well as their dedication to the field of study. These people also tend to be very, very busy. Your response to the 'board' will likely be entered by a junior editorial assistant and it's highly unlikely that the editorial board will even get involved -- except perhaps to solicit names of other reviewers. Quite simply, it's too small of an issue and people are far too busy to get worked up over this. While I've heard of editors being angry at junior scholars for not publishing in their venues, I've never heard of anyone being angry because someone didn't serve as an article reviewer. A declination then is risk-free. But going out of your way to declare "I'm not reviewing because of my political position against ____" is making a statement (which is what you want, right?) and thus carries the risk that that statement is offensive to some.