Moving air = some static electricity
Moving air + dust = even more static electricity
Is is enough static to do harm - that depends on how much air/dust that you are moving around and where the static builds.
Moving air/dust thru a plastic tube (vacuum cleaner) = lots more static electricity. This can be somewhat mitigated if the plastic has been infused with carbon in an attempt to make it conductive, but the user would need to remain grounded and in contact with that plastic/carbon for that to be of much benefit. I've seen some vacuum systems (slightly different that the one in the link in the previous post) that use metal attachments and the entire vac is grounded via the power cord.
With plastic attachments on a vacuum - one danger is that the tip of the attachment (which is nearest the computer parts) will become charged. You really don't want that tip to touch or come close enough to a part to arc.
With a well grounded metal attachment on a vacuum, you still run the risk that the tip will get too close to some computer part and become the discharge point for a static charge. Rapid discharge = danger for chips.
Some say that it is best to leave the desktop plugged in during this cleaning process. That might be best for the desktop - that might not be best for the human - since there is often power to the motherboard that the human can come into contact with. Others say to just ground the case w/o plugging it in (presumably so that any static charges will have a place to go and not continue to build and build and then arc/damage).
One problem with the "just ground the case" method is: the preferred method of getting rid of an static charge is to bleed it off slowly. The ESD control industry goes to great lengths to produce work surfaces that are not too conductive. And they make lots of test equipment to let one check that the surface is just right. Then some geeks come along and thwart that by providing a "good solid earth ground" to the case - thus allowing the most damage to occur should a static arc to (or from) a component happen. Sadly, that same "good solid earth ground" can prevent an arc due to static build up too... so what to do?
Some will argue that if the case is grounded and the vacuum is grounded - then where is the potential (pun intended) for disaster? In theory, the air/dust mixture swirling into the vacuum's attachment is a source of static. Is this a real danger? I cannot say for sure - I do not have an ESD safe vacuum
What I do personally is probably not the safest thing for the computer - it is a compromise, a trade off of the risks involved. I don't ground the case (I probably should). I use very slow air, just fast enough air flow to get the job done. The air is from a garage shop style air compressor. I keep the nozzle about a foot away from any components - except for the power supply grills. I get closer to them.
It is probably better to ground the case (w/o plugging it in)... but that is rarely convenient given the location of our compressed air system. It is probably even better to ground the case using some resistor - but that is impractical for most people.
The air coming from the air hose that I use is also moving thru a plastic hose and the nozzle does not have any special stuff attached to get rid of a static charge. I've measured the results of my cleaning method using the tools that the company that I work for provides and I see almost no static build up.
Can components that are already mounted on a board withstand these levels of static that we are talking about? The typical answer that people give is something like: "I've been doing <insert method here> for years w/o any problems".
Since most chips can still function after being damaged by ESD; it is hard to say what the best method is... opinions vary. But I wouldn't toss the computer in the Rhine for a rinse.
This is already into the realm of TL;DR - but I have one more warning:
Do not use a "regular" vacuum to clean up printer toner