Stone Fruit
Peaches, plums, apricots, nectarines, etc. should be stored at room temperature until ripe. Storing in a paper bag with ethylene-producing fruit such as apples, pears or bananas will hasten ripening. After ripening, store in the refrigerator for up to three days. Storing peaches, nectarines and apricots in the refrigerator will affect quality as the chill robs the fruit of its juice and flavour.
Apples and Pears
Will keep for a few weeks in the refrigerator if stored loosely. Pears need to soften at room temperature prior to eating to ensure ripening. Do not store apples at room temperature.
Source
My dad keeps all fruit stored in the refrigerator, but checks them regularly and never leaves them in there for longer than a week. My mom prefers her fruit to be at room temperature, so she leaves them in a bowl on the counter until she wants to eat them.