Be aware that depending upon the wattage of your hair straightener,
you may find that your dual voltage device will blow the circuit
breaker when you use it.
Thirty years ago, I was staying in a hotel in London, and there was an American style plug in the bathroom. I don't remember if it was 120V or 240V, but my hair dryer was dual voltage, so whatever it was, I used it. My hair dryer ran for about 2 seconds, then quit. I checked the breaker box in the room and the tripped breaker was for 1200 watts (5 or 10 amps, I forget), but my hair dryer drew 1500W. Apparently the American plug in the bathroom was just for shavers, not hair dryers.
I don't think hair straighteners draw 1500W like hair dryers do. The ungrounded two pin Europlug that is so often recommended here for continental Europe is rated for 2½ amps (575W at 230V). Hair straighteners could exceed this wattage, so you shouldn't use them with a Europlug unless you know that the straightener draws less than 575 watts.
Additionally, American hair straighteners have polarized plugs. This is a requirement by UL because the straighteners only have a one pole switch on the line side. A two pole switch necessary to avoid having to to use a polarized plug would cost a few pennies more, so manufacturers go with the polarized plug option, which makes them unsafe in Europe with it's mostly non-polarized power. An internal short could turn on the device even with the switch in the off position. My advice is not to use an American straightener in Europe. Since you won't follow my advice, at least unplug the straightener when not in use.
Swiss power, by the way, is polarized (a grounding plug can only go in one way). Using a grounding plug adapter, like this one, in Switzerland, will preserve the safety of the polarized plug.