Does drinking coffee have negative effects?
First of all, the fact that a substance is beneficial does not mean that our body will, perforce, produce it naturally. Think of oxygen and water for example1.

As for the negative effects of coffee drinking, yes there are many although coffee has both a positive and a detrimental effect on human health. The main active ingredient in coffee is caffeine which can have the following negative effects:


Caffeine can increase blood pressure in non-habitual consumers.
High blood pressure is associated with an increase in strokes, and cerebral 
vascular disease, which in turn increase the risk of multi-infarct dementia.
Caffeine may reduce control of fine motor movements (e.g., producing shaky hands)
Caffeine can stimulate urination.
Caffeine can increase cortisol secretion, some tolerance is developed.
Caffeine can contribute to increased insomnia and sleep latency.
Caffeine withdrawal produces headache, fatigue and decreased alertness.
Caffeine is addictive.
High doses of caffeine (300 mg or higher) can cause anxiety.
High caffeine consumption has been linked to an increase in the likelihood of experiencing auditory hallucinations. A study conducted by the La Trobe University School of Psychological Sciences revealed that as few as five cups of coffee a day could trigger the phenomenon.
High caffeine consumption accelerates bone loss at the spine in elderly postmenopausal women.


The list above comes from the very comprehensive wikipedia page on the health effects of caffeine. In fact, wikipedia is particularly good on this subject, see the following pages:

http://en.wikipedia.org/wiki/Caffeine

http://en.wikipedia.org/wiki/Health_effects_of_caffeine

http://en.wikipedia.org/wiki/Health_effects_of_coffee

1 Our bodies do produce both Oxygen and water as byproducts of various reactions but nowhere near enough to satisfy our need for these substances.