I’m interested. I believe you can change air filter every week and won’t hurt anything to the engine but your wallet.
And not changing it won’t “damage” the engine but affect the engine efficiency and performance.
What we call a 'paper' filter, which is pretty much any factory style air filter, is least efficient at removing dust when it's new. As it gets dirty, that dirt fills in voids/spaces in the filter and while doing so, it will filter more (and smaller) particles. You do get to a point where there's so much dirt on the filter that performance is reduced, but that actually takes quite a bit.
Roughly, in the first ~5% of the filter life, it lets in 1/3 of the total dust over it's useful life. So, if you change the filter out after 5%, it would only take 3 filters to match 1 full life filter. Now repeat that filter change 17 more times to reach that 100% full filter life, you let in almost 7x the amount of dust as 1 full life filter. Those are just rough ballpark numbers, but are based on actual lab results.
One other point is that every time you crack open the air box, you're exposing the clean side of the ducting to environment. Again, the amount is so small probably not noticeable, but still there.
Whether that causes
noticeable damage to an engine is really environment dependent. If you drive 100% paved roads in a city, you probably won't see a noticeable degradation. If that's flipped to 100% dirt roads, you might see degradation quicker. With so many components in the engines that rely on engine air/engine oil, better to keep anything and everything you can out of the intake air.
As for efficiency/performance, you could see a performance decrease at the extreme clogged filter, but the problem is that it's so gradual, you probably don't notice it until you put a new filter in. Efficiency (fuel mileage) takes quite a bit to degrade.
Real life case, the first air filter in my motorcycle ran until the bike wouldn't start at all. It actually ran fine on a Friday, but wouldn't start the following Monday. Fuel efficiency did not change at all over the life because it is fuel injected, so the ECU was able to compensate. Power on the other hand, that's where things were different. The power decrease was so gradual, that while the bike still felt fast (06 GSXR750), when I put the new filter in, it was a completely different animal.
My last comment, we had a customer that was changing the filter out in the off-road construction vehicle every 3 days because it was 'dirty.' Well, the engine wore out (rings) much sooner than expected. Looking at the restriction of the filter after 3 days, there was minimal increase in restriction, so the filter was basically new. Just because the filter looks dirty doesn't mean it really is.