Open the flue of the chimney to allow more air flow. Air circulation gets rid of the liner smell at a faster rate.
Wash the liner by wiping it down with a soft sponge and mild detergent to help remove lubricants and oils from the manufacturing process. You'll be limited in how much of the liner you can wash due to space restrictions and limited reach but even getting some of it helps remove odor.
Position a fan in front of the chimney to blow more air through the liner and up the chimney. This increases the ventilation speed and removes liner odor faster.
Spray the liner with white vinegar from a spray bottle. Vinegar is a deodorizer and the vinegar smell dissipates naturally in a few minutes.
Sprinkle baking soda into the fireplace and around the base of the liner if the smell persists. Baking soda absorbs odor, which is why it frequently gets used in refrigerators.