I honestly don't know what happens if the valve clearances go significantly out of whack for long periods of time (shortened engine life? poor performance?), but clearly Honda means for you to keep them in spec or they wouldn't have put it in the owner's manual. Lots of dealers have been deliberately skipping this important step because it can take lots of extra time for not as much extra money.
My experience comes from Volkswagen beetle engines and Honda Civics. So, for what its' worth...
The valve clearance is necessary to ensure that the valves are fully closed during compression and combustion. The clearance changes as the parts of the engine expand and contract with heat. If the clearance should ever become less than 0.000" the valve will leak and this can cause a lack of power. Far worse though is that the valves, and especially the exhaust valves, are cooled by being in contact with the valve seats that are in the head itself. If they don't close completely, they overheat and burn. Intake valves are less prone to burning.
If the valves get damaged, they can be refaced or replaced. If the valve seats get damaged they might possibly be refaced but more likely the head would need to be replaced. It would probably be better to get a new one unless you can be sure the face angles and widths are done well.
If there is too much clearance, the valve action will be noisy but I am not aware of any particular damage that might occur. The valve timing and lift are affected by the clearance. It used to be popular to run beetle valves with low clearance to 'get more power'.
All that said, I am a bit surprised at the amount of clearance specified as engines with such short valve trains don't really have a lot of parts to expand unlike the old push rod engines.
So, I guess the rule of thumb might be that too much clearance is better than too little.