You can stress an amplifier with a harder load than it is designed for. Speakers, on the other hand, are almost never damaged by impedance itself; they are damaged by clipping from a small amp driven too hard. Keep those two facts separate and the whole topic gets calmer.
The rules, in order.
The amplifier manufacturer's stated supported-impedance range is the boundary. If the manual says 4 to 16 ohms, a nominal 4 ohm speaker is not a mismatch at all, it is inside the specification. If it says 6 to 16, a 4 ohm speaker is outside it, and forum bravado about "it will be fine" does not change what the output stage can deliver.
Why does a lower impedance stress the amp? Because for the same volume it has to supply more current, and current is heat in the output transistors and power supply. A modest AVR asked to drive demanding 4 ohm speakers loud in a big room runs hot, trips protection at volume, and in stubborn cases ages faster. Heat is the tell: if the top of the AVR is uncomfortable to touch after a film, that is the load talking.
The really dangerous case is not a 4 ohm speaker at all. It is wiring two speakers in parallel on one channel to "fill the room" without realising you have halved the load. Two 8 ohm speakers in parallel present 4 ohms; two 4 ohm speakers present 2, which almost no home amplifier will tolerate. This is how amps actually get killed at home, usually in the name of a party.
Also note that nominal impedance is an average; most speakers dip below it at some frequency, often in the bass. A speaker rated 8 ohms with a 3 ohm dip is harder work than the label suggests, which is why "rated for 4 ohms" on the amp side is the safer margin.
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