Does using Activity Monitor have an effect on server performance?
YES. Beware of the "observer effect."

This article is about you using trace and extended events directly, but Activity Monitor uses similar technology under the covers - it doesn't have any magical, no-cost ways of collecting performance information.

There are several monitoring tools out there that do a better job at this than Activity Monitor, some more so than others. There are also free scripts like sp_whoisactive and sp_BlitzFirst that are more honest with you about what's really going on on your servers (see this article). I strongly suggest you try some of those out and don't open Activity Monitor or Profiler at all.