Sometimes when an insert is installed, the installers can leave the flue closed. When you start the fireplace, the exhaust will have nowhere to go and the absence of a draft will create problems. Sometimes the insert may get in the way of the flue so it can only open partially. Other outside events, from a windstorm to a bird nest, can also clog the chimney flue and prevent necessary air flow.
If you install a fireplace insert, the insert may be more airtight than the previous fireplace. Other remodeling can also make windows and doors more resistant to air flow than before. These changes affect the air pressure in your house, essentially removing the possibility of any drafts that actually kept the fireplace "breathing" healthily. You may need to add extra vent room to combat this problem.
Unfortunately, with some inserts the opposite problem can also occur. Sloppily installed inserts can have leaks around the edges of the firebox and chimney. This may be common, since inserts can be too small for chimney spaces. If not properly sealed, the oversized fireplace will not allow a proper draft, and smoke or exhaust can leak out into the room. A smoke guard may solve these problems.
If your chimney has a large open space above the fireplace insert, the exhaust from the fireplace can sometimes settle there more easily. The natural drafts may not be enough to carry it out the chimney as fast as it should move. This can lead to creosote buildup, especially for wood fireplaces, as the smoke lingers and bonds ash particles to the walls of the chimney. Creosote buildup can be a serious fire hazard.