The main problem with using a chainsaw on a zombie is not the flesh or bone, since they are bulk materials that a chainsaw will cut with relative ease.
The main problem is cloth. There is a class of industrial accident where a person gets their clothing caught in powered machinery, and if the machinery in which the clothing gets caught is not razor-sharp, the cloth gets drawn in or wrapped around the moving object.
From there, there are a number of different outcomes. One is that the cloth is drawn tight enough to tear skin and break bones and cause injuries that if they are not an outright amputation, are severe enough to necessitate a surgical amputation. This tends to occur with large, powerful industrial machinery.
The other potential outcome is that sufficient cloth is drawn into the machinery to jam it. When successive layers of cloth are drawn into a small gap by a moving tool, the cloth acts like a brake. This is more likely with a power tool such as a chainsaw
From this it can be seen that the strongest material that a chainsaw must cut when attacking a clothed person - or zombie - is not the person/zombie, but potentially their clothing.
So, the liklihood of a chainsaw to successfully cut through a human, be they alive, dead, or undead, would depend heavily on the maintenance that the tool receives. If it is kept clean and razor sharp, the blade will cut through cloth with ease, but should the blade get blunt, using it against a clothed human is to risk it jamming, binding it to it's intended victim by their clothing.
A chainsaw is not the best possible tool to use against clothed humans. The blade is designed to cut hard, fibrous materials such as wood, and not soft, flexible and strong substances such as flesh and cloth. To cut wood, a channel of sufficient width for the blade to fit within must be cut, else the blade will jam.
To cut flesh, ideally a narrow, sharp blade would be employed this would also make cutting cloth easier. However, a chainsaw blade modified in such a manner would be difficult to achieve, especially in the midst of a zombie outbreak.
Most realistically, a successful wielder of a chainsaw against zombies would keep his weapon meticulously clean and spent a great deal of time keeping it sharp.