Question: A friend of mine who has water-treatment engineering degree recently moved to another country but struggles to find a job in his field. Because he did not want to stay at home unemployed (and because stealing food is not an option), he took a minimum-wage job (salesman in a small shop) that has nothing to do with his skills or even experience.

While I believe there is no shame for him to do that, I'm still not sure what to advise regarding his public LinkedIn profile and/or resume:

Should he mention the low-paying, unrelated job ? Is there any drawback in doing so ? Any drawback in **not** doing so ?

Answer: 
One advantage of mentioning the current unrelated job is that it shows that you are employed in city X. Instead of showing that your current job is in another country, or that your most recent job ended last year. 

It also shows that you have the necessary work permits for the country where you are looking for a job.
Your friend shouldn't mention the low paying, unrelated job on his Linkedin profile. The idea is to keep the Linkedin profile consistent and on point and to send a clear message through that consistency and that relevance.

Your friend can always explain himself at the interview stage. It's not about being ashamed, it's about sending a clear message about the work experience and skills set through consistency and relevance.