Will a fire pit damage my patio's ceiling?
Unless you limit usage to tiny kindling fires, this is a bad plan. Sparks are common when burning wood, and a slight breeze can put them almost anywhere. Logs burn at 600-1000 degrees F or more and hold a lot of heat in case of a tipover, etc.

You will see smoke deposits almost immediately, not "over time". Wood smoke is very dirty. It's also likely that the ceiling will get hot enough to sustain physical damage, especially on calm days.

Then, the notion that wood smoke is somehow less dangerous from a health perspective is simply wrong. It contains many carcinogenic chemicals and other irritants. Having an unvented fire in a confined space and that near your house is simply unwise.

It's almost certainly illegal as well.