You ask "When is insulation necessary?"-- It depends on what you mean by necessary. First, if the closet wall you are talking about is not an outside wall, exposed to the heat(or cold), insulation may not do much of anything except deaden the sound. Even if it is an outside wall, some homes have insulation panels on the outside of the stud walls under the siding that may be sufficient. And lastly, considering most homes have a fair amount of outside walls, your closet may only be a very small part of the total heat gain/loss, especially if the closet door remains closed most of the time. Now it just becomes a matter of cost-- if the cost of the supplies is reasonable, you may want to put fiberglass insulation in there even though the payback might be over a long time...
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