Stress must first be defined, which is not easy. Lets take a mechanical approach to start with.
Stress is a measure of the internal forces acting within a deformable body. Similarly, strain is defined to be a measure of how the object deforms (reacts) to stress. For example, an external force on a spring causes stress in the spring material and the spring strains (stretches) as a result.
Some people view stress to be a measure of the external demands, such as deadlines, placed on a person. If we are to humour the 'spring definition' of stress, this 'demand view' of stress is not quite accurate. Instead, deadlines, relationships, physical activity etc. are more accurately described as external forces/stimuli. I make this distinction intentionally: while it might be good to have many external stimuli, this does not necessarily mean it is good to have a lot of stress.
From a physiological perspective, stress (or strain) can be viewed as the body's reaction to external stimuli which displace the body from its 'desired' equilibrium (homeostasis).
Unfortunately, for many people, meeting the demands placed on them often requires their body to be under stress (ie, their body is strained or displaced from its equilibrium). And for the ambitious people, if their body is not under stress, why not give it more to do, right?
With the distinction between external stimuli and bodily stress out of the way, the effects of stress can be considered. Before listing some examples, a sweeping (yet quite accurate) statement can be made: nearly all effects of stress are bad.
To understand why stress might be a good thing, the effects of stress can be very loosely linked to the effects of cortisol. Cortisol is excreted by the adrenal gland when the body believes it is not in homeostasis. Cortisol causes glucose to be redirected to vital organs such as the brain and heart in order to overcome the difficulty at hand and regain homeostasis (notably, energy is redirected AWAY from the body's immune system). These effects can be considered to be the initial benefit/purpose of stress.
Unfortunately, stress has many many short and long term negative effects on the body. Increased risk of heart disease, insomnia, hair loss and depression are a few.
So in summary, while many external stimuli may be desirable in order to increase productivity, if these stimuli cause stress on the body, a large number of negative side effects will result. A person should strive to achieve all demands placed on them without being stressed.
Final note:this discussion does not necessarily disagrees with other answers; simply, a different definition of stress may have been used.