It really depends on the exact design of the charger, really. A current rating of 1A doesn't necessarily mean that the charger will do any regulation to ensure the output current stays at 1A if the load is too low. It just means that as long as the current stays below 1A, the charger will continuously deliver the desired current at the correct voltage, without overheating or cutting out automatically. Which of the latter happens on higher currents, again, depends on the design of the charger and/or load.
While normally I would expect genuine Apple products to at least not blow up under such a scenario, I do vaguely remember some videos about old Apple chargers being crap, especially the super compact ones. And if you have a counterfeit charger, all bets are off, of course.