Do bikes with carbon frame get detected by induction loops?
The loops actually detect the metal in your wheel rims more than the bike frame:



So it's less about a carbon frame and about having carbon or plastic wheels. 

One trick if the loop doesn't detect your bike is to tilt your bike so that the crank and pedals (which are mostly made of metal) are nearer to the loop and the ground. 

Some areas will paint the road with the location of where the loop is most sensitive to bicycles. 



They also sell magnets to help motorbikes trip automotive loop sensors but I'm not sure of how well they work -- on motorbikes or bicycles. 

Addendum: I have a big honking ebike so I don't have problems tripping the bike specific sensors here in California. In Connecticut where I rode a road bike and the sensors weren't configured for bikes, only cars, I did have more trouble both with my motorbike and my bicycle and I seriously considered getting a magnet.