Generally there are no known health side effects to eating mostly or only freeze-dried food.
Freeze-dried food tends to hold more nutrients, and hold them longer, than most other shelf-life extending approaches. Freeze-dried fruits hold nearly the same amount of vitamins and antioxidants as their fresh equivalent. Freeze-dried food also does not need to have preservatives which have debatable negative effects (depending on type and quantity). Additionally, food preserved by freeze-drying stays nutritious for longer than any other approach (much longer than some).
Many commercial makers of freeze-dried food add a ton of sodium, but that is not part of the freeze-drying process. Some manufacturers are better than others.
Unfortunately for those concerned about the environment, the process of freeze-drying is much more energy intensive than other preservation methods.
American Institute for Cancer Research article on antioxidants
NASA PDF document that describes 98% nutritional equivalent at 20% of the weight