Improper chimney sizing is primarily an issue when venting appliances. A very common example of this is the fireplace insert or woodstove which is vented without a proper liner into the smoke chamber of a typical fireplace.
This type of installation is commonly referred to as a slammer. Slammers are notorious for creating dangerous messes of creosote which can ignite into a chimney fire. Chimney fires are very dangerous things. In addition to path through heat which can ignite the combustibles surrounding or adjacent to a masonry chimney, burning debris can exit the chimney and ignite a home from the roof down.
The way to eliminate the problems created by a slammer type installation is to properly vent all appliances with a liner sized to each appliance. One may also want to consider updating any woodstove or fireplace insert which was not produced in the last 15 years.
During that time incredible strides has been made in woodstove design making available woodstoves which when installed properly, burned properly, and cleaned annually are much cleaner and safer than those which produced larger amounts of creosote in the past.