While I agree with a lot of what has been said, I disagree with one of the fundamental ideas everyone else seems to have--there are certain characteristics and treatments that makes something feel luxury and wealthy. I'm not sure I know them all, in fact I'm sure I don't, but here's what I've been able to identify since posting this question and really thinking about it.
Luxury is often about the craftsmanship and quality of materials. This is a key point to identifying the characteristics. To elaborate, looking at the logos others have posted:
Very, very well crafted typefaces. These are typefaces that are flawlessly balanced, kerned, spaced, everything.
They are generally a solid color often black, sometimes gold, white, silver. There are no extraneous elements. No drop shadows, no outlines, no emboss. They stand on the merit of the type's craftsmanship, again like the underlying product.
Only time rounded corners occur are when used in a signature based identity such as Agnes B. Voyage and Salvatore Ferragamo. This could be extended to say only time letters aren't necessarily crisp, balanced, and kerned are also in signatures and could include Christian Louboutin logo into the mix.
There is rarely a marking associated, and when so it is frequently using the actual type as is the case in Fendi and Chanel (Chanel of course also tributing Coco Chanel). The only luxury brand I can think of that deviates from this is Versace. However, the type used is still in line with everything else. Watches and cars are more open to adding things however.
Everything is on a perfect horizontal... baseline, crossbars, ascent line, cap line, etc..
The main takeaway is that luxury doesn't scream it. Its subtle, sophisticated, and well crafted. It lets the type speak for itself. To further illustrate this lets look at some toy companies:
The type doesn't hold on its own. Spacing isn't as perfect, letters are more playful, angles and curves are used, colors not only around them but in the actual type, then there's shadows, strokes, gradients. As some pointed out the free space in luxury logos --- well its not just the free space but the lack of additional elements. Luxury isn't "minimal" but it is clean. These elements don't appear in luxury. Lots of rounded
Can think of it like food. A high quality restaurant uses the best ingredients and allows it to speak for itself. A typical lower end restuarant would be processed. The Toy companies are processed type, while the Luxury brands are the high end ingredients.
To user568458 joking in comments about Comic Sans being used in a luxury brand. We don't see it because of the playfulness in the characteristics. The curved ends, the tilted letters, the variations in horizontal lines. All of the reasons people hate Comic Sans MS are the same ones that don't lend it to being used as a typeface signifying wealth and luxury.
To scott mentioning Polo Ralph Lauren. Polo is the lowest end of their product line, their luxury brands are Purple Label (highest) and Black Label. Those logos are:
I happened to be reading an article that had this as its final point:
In art as in life, white space is the ultimate luxury. The most recent architectural addition to the Museum of Modern Art in Manhattan is a superb example of the way acres of empty white walls can be used to make a superlative statement On the page, white space works the same way. It signifies that you have plenty of room to spare. It frames images with an aura of accessibility. The less crowded a magazine layout, the more elitist is its attitude.
"The luxury of less."
One of the first magazine art directors to realize that a blank surface could have as much impact as a printed one was Alexey Brodovitch. His Harper’s Bazaar layouts in the 1950s treated white paper as if it was an electromagnetic field, with blocks of type and photographs charged with positive and negative energy.
Because of its upscale connotation, white space was vilified in the 1970s as too elitist. Only in the late 1980s did a handful of art directors, Neville Brody and Fabien Baron among them, dare to reintroduce white space in magazines, chiefly on opening spreads, to show off their typographical wit.
Source: The Object Poster, the Visual Pun, and 3 Other Ideas That Changed Design, The Atlantic, 12 Apr 2012
I think this offers some valuable historic insight into the use of white space by luxury brands.