What you can see from the ground
Brick faces flaking or popping off, mortar joints you could scratch out with a key, sand collecting at the base, a visible lean or bulge, and pieces of concrete in the gutters all indicate water has been getting into the structure for a while.
White powdery efflorescence on the masonry is a related signal: it means water is moving through the brick and depositing salts as it dries.
What you can see indoors
Damp patches or stains on the ceiling or wall near the chimney breast, peeling paint beside the fireplace, and rust on the damper or firebox components all point to water reaching places it should not.
Debris or tile fragments in the firebox is a more serious finding, because it usually means the liner is breaking up.
What you can smell and hear
A tarry smell that peaks on humid days points to creosote. An ammonia or musty smell points to animals. Scratching or fluttering means something is living in the flue and the flue is probably blocked.
Smoke pushing into the room during normal use is the one to act on immediately, because it means combustion products are entering the living space.
The ones people ignore for too long
A stuck damper, a missing cap and a cracked crown are all cheap to address and expensive to leave. Each one is a route for water into the middle of the structure.
A carbon monoxide alarm sounding anywhere near a fuel-burning appliance is not a maintenance item. Stop using the appliance, ventilate, and get it checked.