I often hear that NSAIDS damage the gut, and we all know one of the most common NSAID is Asprin, which is high in salicylates. Do you think this is the component in Asprin that damages the gut? If so, could those new to paleo with continuing gut issues cause further damage by consuming many high salicylates food, specifically coconut products and some vegetables?
Aspirin is not "high in salicylates." Aspirin IS a salicylate (acetylsalicylic acid).
The gut damage due to salicylates and other NSAIDs is due to high concentrations and/or prolonged exposure, which can lead to gastritis and gastric ulcers. Salicylates irritate the lining of the stomach and also interfere with blood platelets, which can lead to gastric bleeding.
With respect to foods, I think you're confusing salicylate toxicity with salicylate allergy. Many foods contain trace amounts of salicylates, which should not cause salicylate toxicity. However, if you're allergic to aspirin/salicylates, these foods can trigger an allergic reaction.
Low doses of salicylates may actually be beneficial in preventing heart attacks and strokes, but YMMV.
Firstly, you're somewhat correct, the acetylsalicylic acid in Aspirin weakens the mucous lining of the stomach and makes it susceptible to it's own acids, this is why you should always take aspirin with food to act as a buffer for this effect.
In the end it's a question of moderation. Anything taken in too high a quantity will cause problems. In a normal, moderately healthy body phenols and salicylates are are easily processed, broken down, and eliminated. As long as you have no natural intolerance to salicylates (and I am dubious of this as most plant foods contain some kind of phenol) then as long as you aren't eating those foods at every meal you're probably not going to have any significant issues.
Keep in mind that all painkillers, including those we identify as being "safe" can hurt the digestive system. Ibuprofen can damage the liver if taken too often.