There are no side products from petroleum distillation.  All parts of a barrel of oil that go through a distillation unit are used to make products.

it really illustrates the kinds of products that come from all parts of crude oil:

I remember seeing a similar diagram in one of my old “big book of science” type books of my youth.  Now that I actually work on refining projects and have a much more intimate knowledge of what goes on in an oil refinery, I know that in reality what’s portrayed here is, as is often the case, a gross oversimplification.  There are a lot of inaccuracies – for example, bitumen is not separated from the residual oil fraction until another set of operations is performed.  The biggest thing that is missing from this diagram is all of the upgrading processes that take things for which there is more supply then demand – things like fuel oil – and turn them into things we want like gasoline or diesel.

One thing that I think this diagram does very well, though, is to point out how many of the fractions you get out of an oil barrel overlap with each other.  To take one example, naphtha fits right into the gasoline range, and in fact this is precisely what most naphtha is used for: a blendstock for gasoline.  Another one is the large portion of the kerosene region that overlaps with middle distillates (in the diagram, diesel oils).  This gives you an idea of the kind of choices faced by refineries in making a barrel of oil into products.

Distillation is in fact only one of many operations used in an oil refinery, although it is the earliest and most fundamental of them all.  If there is truly any unwanted side product from any part of petroleum refining, I would say that it is petroleum coke, a coal-like substance that is dirty and hard to get rid of.  That being said, however, it doesn’t come from distillation but from another operation called delayed coking.