Short answer is yes a MSW inverter can damage some electronics. Typically resistance type loads (hair dryer, plug in power tools, some space heaters) Think items that get hot, but do not have complicated electronics in them are safe for MSW inverters..
As far as the hair straightener. If it was a standard on/off switch and nothing more. Then most likely the MSW did not damage it. If the straightener had a digital display then yes it's possible that it was damaged.
Many electronics will be damaged by MSW. The real challenge is to figure out what electronics will tolerate and what will not.
Another thing to consider. I would bet that 98% of all the inverters on the road are MSW. If you walk into Walmart, Target etc you will only see MSW inverters. Millions of people have them in their cars with their laptops, phones, and kids games plugged in with no damage.
