The gases are much less of a problem than particulates. You could easily imagine drugs (or evolved adaptations) to make people less vulnerable to carbon monoxide, by increasing the oxygen capacity of their blood or by breaking down the carbon monoxide before it gets to the bloodstream.
But humans have absolutely no way of removing insoluble solid particles from our lungs. It's actually something people aren't sufficiently aware of in real life – any kind of fine airborne dust does permanent, cumulative damage to your lungs. Without radically changing the structure of our lungs, the solution is to beef up the limited defenses we already have to prevent the dust getting in there in the first place (nostril hairs and cilia in the respiratory tract). You basically need HEPA filters in your nostrils (and the stronger lungs and bigger airways to be able to breathe through them).
That kind of filter is very difficult to keep clean. We secrete mucus to bind the particulates trapped by our cilia, and then eject the mucus, but this only works for very small amounts of particulates. To live constantly inside an ash cloud, you'd probably need to make far more mucus than people with cystic fibrosis do (and that's already enough to cause serious breathing problems).
The simpler solution used in HVAC systems is to just replace the whole filter as needed. Perhaps your modified soldiers have a cavity where they just plug in replaceable filters, or perhaps they are able to eject and regrow the filters themselves. Like having a big sea sponge living inside your sinuses.
As I say, this would probably mean that their noses and sinus cavities would have to be much bigger – like snouts – and they would need stronger lung muscles (diaphragm and intercostals), which I guess would make them more barrel-chested.
