Does unblocking power bar outlets through short extension cords increase fire risk?
Yes, but negligibly. 

Any connection increases fire risk by increasing the chance of heat buildup due to resistance, sparks due to arcing, etc. The question is how much, and the answer is not much.

Since you already probably have dozens of such connections in your home (including such high-current things as a microwave and kitchen range), and since most of these are low-current devices, I see no reason to be concerned. 

One caveat would be the potential for heat buildup around the transformers. They're normally held captive by wall outlets or power bars and not likely to end up in a heap on the carpet. In combination, many transformers without good cooling airflow could overheat. Don't let that situation arise. 

Make sure all hardware is in good working order and be happy.