It depends on your documents
There is no single rule. Your association's declaration and bylaws decide what is a common element maintained by the association and what belongs to the owner. Read the maintenance section, or ask your property manager for the maintenance responsibility chart many associations publish.
How it often splits
A common pattern: the association handles the exterior, which can include the chase, the siding and sometimes the chase cover and cap, while the owner handles the fireplace, the inside of the flue and yearly sweeps. Many associations do it differently, so treat this as a starting point, not an answer.
Why townhome chimneys need attention
Many Twin Cities townhomes have factory built fireplaces inside sided chases. The galvanized chase covers on top often rust through, water pools and leaks into the framing, and the damage shows up as stains near the fireplace. See chase cover replacement.
What to do next
If you see rust streaks on the chase siding, water stains near the fireplace or a fireplace that has not been checked in years, report it to your association in writing and book an inspection for your unit. A written report with photos makes it clear what is failing and who should fix it.