I found this site on the Internet saying something to that effect:
Warning!!.....Never connect and disconnect HDMI cables and equipment while your devices are powered on (hot plugging cables). The hdmi cables carries a low DC voltage while connected. You will damage your equipment and the HDMI Distribution equipment while hot-plugging cables which will not be replaced under the 1 year manufacturers warranty. Only use the Switched Mode Power supply supplied with the HDMI Equipment since replacing the power adapter might damage your equipment if incorrect voltage is supplied to the equipment.
The Hot-Plug-Detection feature is part of the HDMI protocol. All data-lines are quiet until a hotplug-event is detected via the physical pin 19. A handshake then negotiates a real connection:
The HPD (Hot-Plug-Detect) feature is a communication mechanism between a source and a sink device that makes the source device aware that it has been connected/disconnected to/from the sink device. When an HDMI cable is inserted between the two devices, the resulting hot-plug detection instantiates a start-up communication sequence.
So, unless we're talking about massive static discharge - which should be prevented by opto-isolation if we're talking about a proper implementation - it is safe, even safer than USB which applies 5 V of Vcc right away without any prior negotiation.
Edit: Regarding the physical USB connectors I should point out that the data- and Vcc-/mass connectors are not of the same length. If you plug it in, mass and power is applied before the data lines are physically attached. This works good enough for most applications.
Hot plugging a connector that is designed to be hot pluggable carries minimal risk, but not zero risk. As mentioned above, Electro Static Discharge (ESD) is always a concern, although one of the jobs of a well designed connector is to equalize the potential between both devices before the data lines ever make contact. The Electro Static Discharge Association has more information here: http://www.esda.org/documents/IndustryCouncilWhitePaper3_PII.pdf
You take a risk every time you power on an electronic component. This is true whether it's powered on due to hot plugging, or just by turning on the power switch. This risk is often due to hidden damage to integrated circuits caused by static discharge.
Why is ESD so bad? I assume you have seen an arc welder. From it you should already know that a lot of current driving a big spark can cause steel to melt and deform. ESD is like the arc welder in miniature. A tiny spark that you cannot even see or feel can still be enough to deform a microscopic trace on a chip. The spark does not have to completely "blow the wire apart" and create an immediate failure in order to cause damage. It might be a small spark that damages just a fraction of the trace. It could narrow down the trace smaller than the current it was designed to handle, creating an unintentional fuse.
When a circuit is energized, it undergoes some physical stress, due to the change in energy causing a localized rise in temperature. This causes thermal expansion, which pushes and pulls the traces on the surface of the chip around at an atomic level. If a path on an integrated circuit is already weakened due to some other factor, (a prior static discharge event, previously overheating the circuit, manufacturing tolerances, etc.) it gets weaker every time you energize it. One of the times could eventually be the "straw that breaks the camel's back."
Again, this could be true due to pushing the power button or due to hot plugging. But it turns out it doesn't matter. The important quote from your post is this line: "...the HDMI Distribution equipment while hot-plugging cables which will not be replaced under the 1 year manufacturers warranty." This statement means it doesn't matter if hot plugging is the cause or not. This manufacturer has stated in the contract that the act of hot plugging voids your warranty. The statement does not need to have a solid basis in science to be upheld by a court - it just has to be enough to convince a judge.
Hot-plug as described in the HDMI specification is not the same as plugging in a device while powered on. Hot-plug detect in HDMI terms means that the sync device (TV or monitor) has been turned on and the HDMI socket is now live, i.e. Hot.
Like USB, an HDMI cable can be connected or disconnected from active devices without physical damage to the device. The only concern is that the source might not resend the EDID query to the sync resulting in poor video and/or audio quality.