Would running the stairs negatively impact my lifts for the next day?
Workload and recovery issues pretty much depend on the individual. The easiest way to find out for yourself is to try it, and see how it works. Actually running stairs is harder than a stair machine since you can't cheat by holding on for balance.
I personally have pretty much dropped long slow cardio in favor of interval training, other than an occasional long jog in the park. If a person is reasonably active physically (ie walks for 30 minutes a day at least) 2-4 HIIT sessions a week is enough for heart health.