What items in a house can potentially be damaged by shutting them off at the circuit breaker?
First, the article writer is arm-waving.  A lot.  A device which is destroyed when you sever power is a defective device.   What he's trying to say is that breakers themselves are not made to be used as switches, and particularly, they're not made to interrupt high-current-drawing loads (like your water heater when cycled on).  Except when they are. 

As circuit breakers evolved, they often came out with special variants of the breaker that can do some parlor trick a normal breaker cannot (and are UL-listed for that trick).  The breaker might be rated to also be used in a 208V/3phase panel.  It might be rated for interrupting inductive loads.  Often these breakers are expensive only because of the sunk engineering costs and that they are oddball -- the manufacturers know if they made every breaker that way, it would add almost nothing to breaker cost.  So after a time, they often do exactly that.  And often UL "levels the playing field" by changing breaker specs so all breakers must be that.  

UL (Underwriters Laboratories) tests products for safety, and lists approved products.  The need to use listed electrical products comes up a lot.  

Here, the relevant trick is called an "SWD" breaker.  This is made for a very common habit in factories and shops of using the breaker as a light switch. There is nothing wrong with this or the AHJ would not allow it.  Being commercial they tend to control fluorescent or HID (sodium, metal halide, mercury) lighting.  The SWD breaker is made to be used daily as a switch, tested for many more on-off cycles. The HID breaker is built extra tough to cope with the poor power factor of many HID lights.  Many manufacturers simply make all their breakers SWD rated.  Whether UL is behind this I don't know. 

So likely your breakers are SWD, and rated for interrupting a big inductive load like an A/C unit all day everyday.  Your old breakers, maybe not so much, that's what the article is trying to warn about.  But even then, this is a bit of hysteria; UL tests even non-SWD breakers for thousands of cycles.