Skimmers add oxygen to the water and remove many nutrients both good and bad, so yes you can over skim a tank.
Can you have a tank without a skimmer? Of course...
Many biosystems do better without one because of their need for a higher nutrient count, however the common overstocked reef environment or massive bioload fish only should have a good quality skimmer.
Back to over skimming; when dealing with tanks larger than a Nano and with a moderate to high bioload, it is quite difficult to over skim unless you spend an insane amount of money (50gal tank = skimmer for a 400 gallon).
Ed