Does increasing the air density change always has an effect of a partly submerged object in liquid system?

When part of the object is in the air above the fluid, then there is a "buoyancy" due to the air density. As the air becomes more dense, the effect is to provide additional buoyancy. In the limit where the air is as dense as the liquid, the object (if it floats in liquid) would keep rising to "find the surface".

In response to the last part of your question - "fluid displaced" includes the gas (which is a low density fluid, and normally can be ignored).