Is not using soap when washing cast iron pans safe? Most advice on cast iron pan care says that most of the time you shouldn't use soap when washing a pan. However, that may leave some residual fat/grease from your cooking on the pan. 

It can be fat/oil with a low smoke point which is going to smoke then next time you heat the pan.

Would it be safe to cook in such smoking pan? Should you burn it till it stops smoking? Is this smoke in the kitchen safe for your lungs?
That residual fat is leftover from what you cooked in there before, so it would only be an issue after cooking something with a very low smoke point, washing the pan,  then cooking something with a high smoke point.

Wiping the pan with a paper towel will get rid of the vast majority of this, and if you dry it over heat you can carry on to cook off the last little bit, like a sort of half-baked or top-up seasoning. Doing this before storage gives the smoke smell time to clear before you next use it, but this should be minimal if you cook it off gently. I recommend turning on the cooker hood extractor if you have one. 

I like the heat-retention of cast iron but I've gone over to enamelled (le Creuset). I still use a carbon steel wok that needs reseasoning after using the traditional metal spatula, so I actually rinse, dry with heat, wipe over with oil and heat.