Does the layer thickness have any effect of the strength of the 3D printed object?
3D Matter has published an excellent article on the subject. They find that thicker layers result in a stronger part, with 0.3mm layers giving a part that is around 24% stronger than the same part printed with 0.1mm layers.
One small issue with this study is that it did not look at the effects of temperature. Raising the temperature generally results in stronger parts because the layers will fuse better. It is possible that you could make a 0.1mm part just as strong as a 0.3mm part by raising the printing temperature.
Another consideration for inter-layer bonding is how much the next layer is "squished" onto the previous one. Using a wider extrusion width will improve strength.
The main issue with the strength of FDM parts is that they tend to break much easier along the layers, much like how wood is much stronger across the grain. You have to take this into account when making your design, and ensure that features that will be subjected to stress/strain are printed in the XY-plane.
Much of the difference in strength by layer height can be addressed by printing hotter with lower layer heights. A strand of half the height cools twice as fast, give or take. Faster cooling means weaker prints.
@RyanCarlyle: does that also mean printing in a heated chamber increases part strength?