What are common downsides to leasing an aircraft to a flying club?
A leaseback can be advantageous, especially when you use the aircraft for your own flight training and travel.  But there are some things you need to watch out for:


Other people will be using your aircraft - whether people admit it or not, nobody will take care of the plane as well as you do. If you think you might take issue with trash in the cockpit, a scratch on the panel, or a tear in the seat, you might want to give this a second thought. In my experience, flying clubs seem to have pilots that take better care of the aircraft, since membership is controlled. 
You'll have to schedule your flights - just like everyone else (though some concessions are usually made for the owner). The bottom line is, you won't be able to just get up and go fly somewhere at a moment's notice anymore.
You have to play by someone else's maintenance rules - While you have to comply with FARs  in any case, the lessee typically will have stipulations that they (or their designee) will perform any required maintenance on your behalf and your say in the matter will be limited. Of course, you still have the honor of paying for any maintenance performed (typically at little or no discount). 
Your capital is tied up - This can be a good or bad thing. It's good in the sense that aircraft tend to be a good hedge against inflation risk. It's bad because you're precluded from using the money on something else. This is one of the main reasons that flight schools/clubs LOVE leasebacks.
If you're looking to make money... it won't happen - most leasebacks only help offset the expense of operating the aircraft. If you're looking for a return on your investment, better to look somewhere else. Do it only to reduce the cost of ownership on something you want anyway.
Finally, YOU take on the financial risk of the aircraft - in most leaseback situations, you are reimbursed for any use, but are still responsible for any finance and indirect (and sometimes direct) maintenance costs. These have to be paid, whether the aircraft is flying or not. And if business starts to slow down... well, I'll give you a hint: it's not the flying club that has to pay to keep your plane airworthy.  This is the main other reason flight schools/clubs LOVE leasebacks.