What would happens if due to increased volume and reduced density, the global warming cause consistent loss into space of certain gases of atmosphere? What would we observe?
Depends on what kind of warming we are talking about.
There is one handy chart: chart
The Earth needs to get quite warmer before it begins to lose water. However, water molecules can dissociate, and hydrogen would readily escape into space. This process is going on throughout the Earth history, but it's quite slow. Without heating, oceans should last for at least another 5 billion years.
Eventually, due to warming caused by increased solar luminosity, Earth may lose most of its water and turn into Venus - but even if we would get to water boiling point (100C at 1 atm pressure), losing all water would take millions of years. Other gases - nitrogen, oxygen, carbon dioxide - are likely to stick around for billions of years.
Firstly this is not even a remotely likely scenario. However if the rate that gases leak from the atmosphere into space was dramatically accelerated for whatever reason, it would eventually have serious consequences. There would be less protection from cosmic rays and solar storms leading to increased rates of cancer.
The atmospheric pressure would decrease and would be able to hold less moisture which in turn would affect the climate and the amount of rain fall.
A reduced partial pressure of oxygen would have adverse consequences for all animal life, causing stress on animals that needed the most oxygen first for example animals that need to run fast. Eventually all animal life would die due to lack of oxygen and all plants as well as plants also need to respire.