I drink two cups of ginger tea daily. Should I switch to coffee since ginger is a natural inflammatory?

I have not found that avoiding any consumed products affect the results of CoolSculpting, so in my opinion, you would not need to stop drinking ginger tea if you were to have a CoolSculpting treatment.  

You can continue drinking ginger tea.  It will not affect your results. 
Swelling is a side effect after Coolsculpting.  When fat cells freeze, they die and are then naturally eliminated through the body’s natural process.  As part of recovery, your body will bring additional blood flow and fluids to the treated area, which may lead to swelling in the area(s) that you’ve had treated.  
Ginger tea is a natural anti-inflammatory that can help reduce your swelling.  Bromelain and Arnica are other great products to reduce inflammation.  They also reduce pain and bruising.

Hello there,
     I do not think this will affect the coolsculpting in any way. Ginger is healthy for the body and would not be strong enough to be a contraindication. Hope this helps.
Sarah RN BSN