Should particle board treads be removed before finishing my stairs?
Purpose-built bullnose 1-1/4" particle board treads are a common and perfectly acceptable permanent tread solution. They've been the standard under carpeting in my region since at least the 1980s. 



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The only time they're a concern is when there's significant moisture contact, but then there are usually other problems of larger magnitude. 

Now if you're asking because you're putting in full-thickness (1" or better) hardwood treads, then yes, you'd remove those.