Well even for knives with no wood, a dishwasher is a very hostile environment.
The reason is primarily for the blades.
If you have quality knives that you care for, and plan to keep for many years, then it's just not worth it.
It's just too easy for a knife to be jostled around and bang into other knives or silverware and get nicked.
You mention that you have a greater chance of getting cut when washing by hand.
Well, there's a trick to that.
The easiest and safest way to wash a knife by hand is to press it flat against the side of the sink, then use your sponge/scrubber on the exposed side of the knife.
Repeat for the other side.
This keeps your blade safe, because the edge never touches anything.
It also keeps your sponges and hands from being slashed to pieces by the blade.
Dry it immediately with a dish towel, using a pinching swipe from bolster to tip from the back of the knife .
cooking.stackexchange.com/users/60/hobodave
Expensive knives can survive a trip through the dishwasher, but like others have mentioned, they can bump into things, end up with coatings of detergents and such, suffer damage to wood, etc.
Why would you do that to an investment?
Cheaper knives will just straight out
rust/corrode, even if you remove them and dry them right after the cycle.
Yeah, you can scrub it off, but you're also losing metal from the blade, possibly starting rust in areas you can't dry, and generally degrading the knives.
cooking.stackexchange.com/users/832/melle
The biggest reason I've heard is that the wood handles can't survive in there and still look good.
It can still work, but you have to be vigilant to remove the knives immediately on completion of the wash cycle.
And yes, it's true.
My wood-handled knives are noticeably grayer than their newer brethren.
Our non-wood handled knives (those Wüsthof blades) go in the dishwasher with nary a worry.
cooking.stackexchange.com/users/798/sysadmin1138
I've heard that certain components of dishwasher detergents may actually be so agressive you run the risk of making your knives dull.
I'm not an expert on metal <> cleaning agent interaction, though, so kindly don't take that as an absolute truth.
What I have observed is that just putting the knives in the dishwasher loosely (as you oftentimes have to do due to the length of the knives) tends to have them rolling about a bit during the cleaning process, which makes them bump into other (metallic) objects such as pots and pans.
That will make them dull, and that is why I always wash my proper knives by hand.
cooking.stackexchange.com/users/744/Boetsj
The easiest way to damage any metal edge is to expose it to water and chemicals for longer than necessary.
When the water just sits on the blade, the corrosion causes the metal to flake off (at a microscopic level).
Immediately wash and dry your knife after use, and it will last you the rest of your life with minimal damage.
I have also heard of some folks brushing rubbing alcohol along the blade to help dissipate the water.
This goes for shaving razors as well.
cooking.stackexchange.com/users/1612/sledge
Adding this answer to provide some (admittedly anecdotal) evidence: I currently own a knife that was given to me by someone else.
The previous owner regularly put this knife in the dishwasher.
It is not a cheap knife, but from a well-regarded consumer brand (Zwilling-Henckels) I would guess it is of medium to good quality.
Two things have happened to this knife: On the handle, water got in between the tang and the resin scales.
In time, there was a buildup of corrosion, which split the resin scales.
The blade itself has developed some pitting (difficult to see in the picture, it is mainly on the spine, more clear near the tip.)
There was also some damage on the knife edge, but as the previous owner also used a cheap draw-through sharpener, it is more likely that that caused the edge damage (it required complete reprofiling)
Based on this, I feel sure that always cleaning my knives by hand is the best way to treat them well.
(That, and not using cheap sharpeners)
cooking.stackexchange.com/users/51657/Anique
The stainless steel used for edges is not the same kind of stainless steel that is used for pots and bowls; it can be hardened far better but is by far not as corrosion resistant.
Look up 300 vs 400 series of stainless steels.
No one would usually make a sharp knife out of 300 series unless they are making a dive knife.
Also, especially expensive knives are not always made out of stainless steel at all, there are plenty of examples made out of non-stainless carbon steels or so called semi-stainless alloys (for example D2 aka SKD11).
Some of the cheapest blades are the most corrosion proof.
Dishwasher chemistry is harmful to all of these.
Also, thermal cycling, especially if uneven, can cause things to warp; with some blade designs warping is very bad news since they will be difficult to straighten and difficult to sharpen cleanly while bent.
Of course, handles made with wood, lacquer, bone, horn... are not dishwasher proof anyway.
cooking.stackexchange.com/users/35312/rackandboneman
the heat and water absolutely destroy wood handles.
i can't think of a faster way of ruining a good knife with a wood handle.
dishwashing agents also corrode metal - including stainless steel.
i've had good knives get pitted from bits of dishwashing powder stuck to it.
the problem is especially bad for carbon steel knives.
don't do it - wash by hand.
cooking.stackexchange.com/users/4179/Chef
My knives have wooden handles which are not dishwasher friendly (the varnish has a habit of melting).
I'm also suspicious of the effect of the rinse aid which seems to add a thin coating onto everything, which might not be good for the blades.
cooking.stackexchange.com/users/771/Rowland Shaw
I don't see any mention of the fact that the heat/cooling causes small amounts of expansion and contraction.
Over time, this weakens the seals between the handle scales (plastic, wood, pakkawood, all of it) and will result in handle scales separating from the knife.
Not the only reason, but worth including.
cooking.stackexchange.com/users/53137/J K
What is it about the dishwasher that harms knives?  The only reasons I've found are:   You could get cut... I personally have a greater change of getting cut by trying to wash them by hand.  The blades can damage the plastic-coated metal shelves... so I put them in the silverware-holder instead of on the rack (it's 100% plastic, no metal).  The blade could get nicked by being knocked into other silverware... so put the blade by itself in a section of the silverware-holder in the DW.
Why not put knives in the dishwasher?
cooking.stackexchange.com
<equipment><cleaning><knives>
cooking.stackexchange.com/questions/1804
cooking.stackexchange.com/users/364/JustRightMenus
Knives can be jostled around and cause damage to themselves or other dishes. The handle can be damaged by prolonged exposure to water and cleaning products, and the steel blade can be coated with detergent or become dull or corroded  A good knife should not be washed in a dishwasher as it will damage the blade over time. A wooden handle will also suffer badly from the heat and water.
Washing in a dishwasher runs the risk of knives being damaged by other items. The wooden handles are not designed to go in a dishwasher. The chemical detergents in dishwashers will damage the knife blade.