Why should some piston engines not be run lean of peak?
Considering piston engines in general there's a few reasons not to run "lean of peak", but the arguments I'm aware of all fall into three broad categories:


Poor Instrumentation
Most folks advocating Lean-of-Peak engine operation will also tell you that you need a good engine monitor (with Exhaust Gas Temperature (EGT) and Cylinder Head Temperature (CHT) probes for each cylinder) so you can ensure that all your cylinders are operating within safe temperature limits.
Manufacturer CHT/EGT systems are often "single-probe". On the IO-360 in your Cessna example that leaves you with a 3 cylinder blind spot in your monitoring, and the potential risk of damaging your engine based on the incomplete information you have.
Uneven mixture distribution
Some engines simply can't run lean of peak because of uneven mixture distribution: Either the engine will get rough as some cylinders "miss", or you'll have some cylinders lean of peak EGT and at a safe cylinder head temperature while others are redlining their CHT and potentially causing damage to your engine.
This sort of uneven mixture distribution happens frequently in carbureted engines, but there are many injected engines which also have poor fuel distribution &amp; significant variation in mixture between cylinders (This is where GAMIJectors makes their money - by providing balanced injectors to even out fuel distribution).
"In Case Something Breaks"
Hypothetical scenario: you are cruising at 8000 feet, you've leaned for cruise, and your mixture control suddenly fails in such a way that your engine is stuck at the current mixture setting.  All the airports within your endurance range are roughly at sea level.
An engine running lean of peak may begin to run rough as you descend for landing (the mixture becomes too lean to support combustion).  An engine running rich of peak may continue to run acceptably through the descent. (The mixture may transition into "the red box zone" where operation at high power settings are dangerous, but you'll theoretically be reducing power for your descent, so the chance of doing damage to your engine isn't that high unless you have to perform a go-around or other high-power maneuver.




A prohibition on Lean-of-Peak operation in your POH is probably historically based on #1 or #2 above -- the manufacturer doesn't want to advise you to do something that might damage your engine.  Running Rich-of-Peak is conservative operating guidance, where the downside is slightly increased fuel consumption and the possibility of fouling a spark plug (which is not likely at normal operating power settings) versus the downside of potential engine damage from improper Lean-of-Peak operation.