"Cannot deal X" modifiers nullifies damage of X type at every stage after conversion. This means if you cannot deal fire damage, your ignites will be zeroed out and no longer apply.

A more subtle consequence is that if your chaos damage can ignite but you cannot deal chaos damage, that chaos damage will also be zeroed out when calculating ignite damage. This can be tested by allocating Avatar of Fire and attempting to ignite with pure chaos Hexblast, you'll find that you can't despite chaos damage being able to ignite. How this relates to Elemental Hit is that when EH selects fire damage, the only damage that gets factored into the ignites is fire even if your lightning, cold, or chaos damage can also ignite.