Do filtered colimits commute with finite limits in the category of pointed sets?

Yes, filtered colimits commute with finite limits in the category of pointed sets. This is because the forgetful functor from the category of pointed sets to the category of sets creates finite limits and filtered colimits -- in fact, it creates all limits and all connected colimits -- and so the category of pointed sets inherits this property from the category of sets.

I find it sometimes useful to remember that the stated commutation property holds in any ind-category. Pointed sets is ind-(finite pointed sets). Of course, to verify your particular case by hand, working with elements, literally takes 5 minutes. But who wants to waste five minutes on elements?