Prime is a reducing agent. Formaldehyde, an ingredient in Ich-X can also behave as a reducing agent. In the aquarium, these chemicals can "donate" electrons to free oxygen gas in the aquarium water, producing water, hydrogen peroxide, and / or hydroxide ions. In any case, these chemicals lower the overall quantity of available oxygen within the aquarium. Prime and Ich-X likely don't interact directly, but using both simultaneously can cause a larger decrease in dissolved oxygen levels. The caveat here, though, is that when Prime is dosed appropriately, very little should remain in the system in the long-run. It'd also be ideal to use Prime and Ich-X at different times; water change in the morning, medication at night, for example.
So, basically: yes, technically Prime and Ich-X can have minor negative impacts on the aquarium. In reality, however, proper use of these chemicals shouldn't lead to major problems; water changes are more important than a small reduction in oxygen pressure. It's not a bad idea to run an airstone during treatment, though, especially as warmer water (which is desirable for ich treatment) retains less oxygen.