Nothing wrong with dry-firing a center fire cartridge weapon (9mm, 5.56,.308 etc)... in MODERN weapons. Older firearms may have issues due to advancemrnts in metals over the years. However, there may be exceptions that are dependant to certain manufacturers (I don't know of any).
It will damage rim fire cartridge weapons (.22 etc) over time from what I've been told... I don't own any rim fires so I cannot deny or confirm.
I guarantee some will disagree... but every expert in the law enforcement & Marine Corp/Sniper community agrees it causes no harm to the firing pin or action. I regularly dryfire my custom .308 at least 100 times a training session and do the same with my Beretta's & Sig's.
If you really want to play it safe you can use snap-caps... which are great for staging malfunctions and other drills.