This question came up when I was talking about the atmosphere. Someone had mistakenly claimed that as temperature increases, the density of the atmosphere should increase as well. I reasoned from the ideal gas law that, as temperature ($T$) increases, then so should $V$, leading to lower density. But I realized that pressure could increase instead; the atmosphere is not in a sturdy laboratory container where volume is a constant. Under these circumstances, how can one predict whether pressure or volume (or both) will increase?
I'm guessing you mean for a fixed n moles of ideal gas floating freely in the atmosphere. I also assume a short term heat injection (not a source that holds a particular temperature).
Temperature is directly related to kinetic energy. So the first response of the free floating ideal gas will be to become more excited. This will create a slow pressure wave and the volume will expand in response to the increased molecular collisions (the pressure). Of course the volume expansion will leave a slightly low pressure region, and the volume will contract to fill it. And so on until the system reaches equilibrium at which point the molecules will have mixed with the surrounding atmosphere and a measurement of its volume is now meaningless (or at least, must be more carefully defined).