You won't feel the brick change temps esp. if there is a flue liner...
1.The idea is to air seal the sides of the chimney to stop your (already conditioned) room air from going up and out. (save money)http://www.finehomebuilding.com/PDF/Free/021105092.pdf
2. It will stop a fire at each level (required per building codes minimum) instead of a basement/ground floor fire going directly to the attic, burning it down while the main fire is burning up; with you (hopefully not) in the middle. (save money)
3. It will stop contributing to the natural "stack effect" (save money); http://www.wag-aic.org/1999/WAG_99_baker.pdf
4. It will stop damp, humid basement/crawlspace air from wetting the roof framing, rotting, forcing replacement (save money)
5. If someone at a later date (after you move) adds another appliance, possibly wood-burning- to exhaust the gasses, it will be "to Code"- if all else is in order. (save money)
6. It will stop any convective loops next to the chimney, robbing your insulation of R-value.(save money)
7. It will stop rodents from basement/crawl to attic, to nest in your new insulation that you now need to pay an exterminator to remove. (save money)
Good advice by HomeSealed! Did I mention "save money"? Lol.
Page 16, #8--- pp. 19; http://apps1.eere.energy.gov/buildin...ing_report.pdf