MC33063A Boost Converter IC Getting Really Hot and Voltage Sagging
If you have 11V out with 1.5A load current, that's 16.5W.  Allowing for reasonable efficiency of say 80% that's 20.6W of input power.

So with 3V in, you will draw 6.88 A from the input supply, and that will be the average inductor current.  The peak inductor current will be higher by 1/2 of the ripple current. (In a boost, the inductor carries the input current.)

So an inductor with a 2.2A saturation rating is well undersized for your application.

If you had 30% ripple current, then the average inductor current could be 1.87A before saturation.  This would give you 5.61W of input power capability at Vin minimum of 3V, and 400mA of output current at 11V Vout.  (80% efficiency)

You can probably get more out of your converter if the inductor soft-saturates, but as you noted things will start to get warm....