An example of chemical and another one of a mechanical work done by the cells in the body.
Well, it helps to understand what work is. Work is the transfer of energy through some sort of transfer system. So mechanical work could be if you raise your arm straight up in the air. Of course, we are talking about more than cells here, but you get the general idea. If you want something on the cellular level, how about the heat that is generated as a result of cellular respiration going on? The 98.6 degrees Fahrenheit temperature a healthy human being normally has is the result of the sum total of all the cells in your body transferring that quantity of energy as heat energy.
As far as chemical work, what about the digestive system? Let's take carbohydrates, for example, one of the three food types. When you eat a piece of bread, for example, their is an enzyme in your saliva called amylase, which immediately starts the digestion of the bread while it is still in your mouth. This would be chemical digestion, cutting apart the longer chains of carbohydrate molecules into shorter increments that would be available for absorption later in the small intestine.