That's one option, but another would be taxes: increase landing tax for flights, hotel tax, car rental tax, tourist services tax, etc.
Or simply limit numbers: require that people buy tickets to enter particular places (including national parks), and restrict entry once the place is "full".
If Icelanders aren’t happy with their current level of sovereignty, then perhaps they should do that. Otherwise they may consider directing some of the 10% of their GDP they generate through tourism to solving some of these problems.
No, Ireland isn't in the Schengen zone. Irish people are free to visit Europe (since part of the EU). Also a Schengen VISA DOES NOT allow one to visit oversea French territories. You can't visit The Guadeloupe with a Schengen VISA. So things are a bit more complicated and wouldn't necessary impact Icelander's ability to travel to Europe, it entirely depends on what kind of deal Island signed with the rest of the EU.