Heating of mixture of gases with one gas barely above its boiling point What happens when mixture of two gases is heated in a closed container?

If one gas expands faster than the other and increases pressure of the container enough so that the boiling point of the other gas is decreased below the gas temperature, will second gas start turning into liquid?

This phase change must also give out heat, causing the overall temperature and pressure to further increase.

Is this valid? or mixture of two gases work lot different than two of them separately?
You could certainly have such a situation; that is, heating two compounds in their gaseous state at constant volume such that that the total pressure of the system exceeds the vapor pressure of the lower boiling compound causing it to condense.  

As the gas condenses, it will of course give off heat as you said, but it would require continued heat input to sustain the increasing pressure and condensation. I believe that is what you meant in your second question anyway.