Provided that you are healthy, not suffering from a cold, disease state or other suppression of the immune system, moderate exercise during fasting periods should not impact your health. For submaximal efforts, you may notice earlier fatigue, and there will be definite performance impacts the closer you get to maximal efforts.
Caveat: You are still introducing an artificial state (fasted), which may have health complications. I would schedule a consult with a physician and explain your plans, and have them give you a checkup with that in mind before starting this.
However, there will be some impact on the effective level of the exercise, and how well you are able to perform the exercise during the later parts of the fasting period. These changes include depression in max VO2 (Although this value is more of a performance predictor, not really a measurement metric), higher levels of free fatty acids (FFA's) in the blood, as well as a somewhat suppressed gluconeogenesis as well as a concurrent rise in fat based oxidation for energy.
What this means, is that your body will (mostly) use up all muscle and hepatic (liver) glycogen storage, and will turn to fax oxidation (ketone bodies) for fuel. There is not that much difference in the lactate/pyruvate levels during this time, suggesting that the switch is adequate to fuel performing muscles. There are other short term changes in several hormone and substrate (glycogen, pyruvate, lactate, etc) levels, but these are returned to normal after resuming normal dietary intake.
While not an ideal state to pursue, intermittent periods of short fasting ( < 12 hours) or even up to a few days shouldn't have any health impacts, although you may notice some performance impairment.
This study I was able to obtain, only the first page is freely available. This took 5 obese subjects, and in a clinical setting (hospital), underwent a 3-5 week fast, with only water and vitamin supplements for health. They showed the changes described above in hormone and substrates, with a drop in VO2 max levels at the 2 and 4 week exercise marks. There was also substantial weight loss, but that is to be expected. They showed no other adverse health effects during the prolonged fast, but to emphasize, they had adequate water intake and daily vitamin and potassium supplementation.
Another paper available through Dartmouth Sports looked at 12 hour fasting for Ramadan and the effect on athletic performance, and found that even in the limited fasting state, there is some performance impact. One passage stands out in the paper:
An extensive review of the older literature on the effects of
fasting on endurance performance was published by Aragón-
Vargas.(21)
The conclusion of this review was that a short period
(24 h to 4 days) of fasting in humans resulted in a decreased
capacity to perform endurance exercise. In spite of a rather
consistent effect of fasting, however, there was no clear evi-
dence as to the mechanism responsible for the earlier onset of
fatigue.
That is corroborated by other sections in the paper, one detailing performance impacts on cycling at 100% VO2 max level after 24 hours fast, and impacts on high intensity events such as the 100m and 800m runs. Interestingly, one section shows no difference in muscle glycogen stores in the absence of exercise during fasting.
There may be some effect of a reduction in the muscle
glycogen store on the maximal rates of muscle glycogenol-
ysis with a consequent loss of exercise performance dur-
ing high-intensity exercise, but a few days of fasting in
the absence of exercise has little effect on muscle glycogen
content.(18)
They do postulate that because of the metabolic acidosis that starts rising with prolonged fast is part of the reason for early fatigue in exercise, which is interesting.
The second paper also has 57 cited studies relating to exercise in a fasted state.
