What body systems are affected by obesity?
All of them.
Obesity is a condition that affects the entire organism - the effects seen in each organ system are specific to that organ system.
Just to provide a couple of examples, the endocrine system which governs (among other things) body metabolism releases different concentrations of hormones in people who are obese.
The skeletons of obese people are thicker and heavier than those of normal weighted people because the bones have to hold more weight, and underneath the fat obese people tend to be better muscled than non-obese people because performing basic daily chores is a weight lifting exercise for them.
There are obvious effects seen in the circulatory systems, cardio-pulmonary systems, renal systems, gastrointestinal systems... Pretty much if it's in the body, obesity affects it.