(to paraphrase a little...)
I'm currently looking for work in Field A. I also have experience in Field B, but is completely irrelevant to Field A. Should I include both experiences on my LinkedIn account?
It doesn't matter. Showing just your experience in Field A would make for clear, succinct profile for a hiring manager looking for just that skills, specifically. But, people change jobs/careers so often these days that it's standard for your LinkedIn profile to be more of a CV than a resume: i.e. it's for listing everything you've done, professionally, and not just your qualifications for one specific position. Hiring managers shouldn't see anything odd in a LinkedIn profile that lists all your professional activity. At the very least, your experience in Field B could help you show generic leadership / communication skills, and fill any apparent time gaps (for hiring managers that still look for those).
Note, however, that for the related question...
I'm looking for a job. I have experience in a couple different areas, and I'd love to work in any of them again. Should I put everything on my LinkedIn profile?
The answer is (a definite): Yes! Again, your LinkedIn profile is more of a CV than a resume. You want to draw interest from all those fields, so list everything.