It does increase efficiency somewhat, though depending on how you do it it might not increase efficiency enough to make it worthwhile.
An important thing to note, though, is that if you assign dwarves to a burrow through the burrow menu (as opposed to with an alert), you can assign each dwarf to more than one burrow. For instance, you might have one burrow that contained the metal stockpiles and forges, a second that contained the bedrooms near the forges, and a third that contained the central food/drink/meeting areas, assign your smith to all 3, and they'd be able to do jobs in all 3 burrows. This means that you don't really need separate food or drink stockpiles for each burrow, since you could make a burrow that contained the main ones and just assign everyone that is assigned to any burrow to that burrow.
Also note that alert burrow restrictions and normal burrow restrictions work differently. First of all, alert restrictions supersede normal ones. This means that if your emergency shelter burrow doesn't include someone's workspace, they'll stop work as long as the alert is on. Second, alert restrictions demand that your dwarves all get inside the burrow, but normal restrictions just don't allow assigned dwarves to take jobs or use items outside the burrow (unless the job is eating or drinking and they're starving or dehydrated). This means that idle dwarves can go outside their burrow, and no problems are caused if non-alert burrows aren't contiguous. Since dwarves often choose to idle where their last job was, they'll sometimes choose to go to unsafe locations, even if burrowed.
One more thing: word of Toady is that workshops do not generate noise, so that's not a concern.