it’s safe to run mysqltuner on production server
Most monitoring services have small enough impact that they can (and should) be run on production servers.


mysqltuner -- once in a while.  (Tuning is not something that needs repeating very often.)
SHOW GLOBAL STATUS -- periodically.  (Some services use this to build graphs.)
Have the slowlog turned on (continually) even with a low, but non-zero, long_query_time.  (This is my favorite tool for getting useful clues about performance problems.)
performance_schema (early versions had a noticeable impact)
But not the general log -- This is likely to fill up the disk too fast.  When necessary, use it only briefly.


Here's what I request:  http://mysql.rjweb.org/doc.php/mysql_analysis  It includes


one-time SHOW VARIABLES and SHOW GLOBAL STATUS, then run that against about 200 checks.
one-time slowlog digest (after it has been running at least a day) -- followed by an analysis of the "worst" queries.  This often leads to improving the indexes and/or re-formulating some queries.  It may even lead to architectural changes to get away from problematical design patterns.