Is rotted wood used in the construction of a dresser going to cause issues in the future?

That's not rot, it appears to be a small section of 'live edge', i.e. the original surface of the tree, or possibly the former site of a large, irregular knot which is no longer present. Either way it's basically nothing to worry about.

Since the wood appears to be maple¹ this could also be a bark inclusion², which is something maples are known for more than some other species.

It may seem careless or slapdash to leave something like this without doing something (I guess the most obvious thought would be filling it) but you tell us it's on the back, and as odd as it sounds there's actually a long tradition in furniture making of using rougher wood, and even pieces with defects, for surfaces that aren't usually seen. Even some quite fine furniture from the past could have very rough boards hiding at the back, or inside the carcass. Now, as then, it makes good economic sense to use wood that isn't suitable for show surfaces for parts that aren't visible in normal use.

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¹ Quite characteristic to experienced eyes once it has finish on it.

² What happens is that sometime during the growth of the tree something happens that causes a small section of bark to be left behind and completely encapsulated it the growing wood. If the tree is sizeable when felled these bark inclusions can be very deep inside the trunk.