Does Intel test their silicon on the wafer or after the cores are cut out?
Both.
Do they use a set voltage and ramp up clockspeed to a failure of some sort?
Sort of - there are various test points. A test program which sets up the voltage and ramps up either voltage or frequency or both is called a shmoo (don't ask... that's the name
http://en.wikipedia.org/wiki/Shmoo_plot). They take a while to run and testers are expensive and fast testing programs are created which hit the main areas quickly without testing the intermediate points and this is a lot faster than doing shmoos.
Do they test at various voltages and frequencies?
Yes.
What type of application do they use to test silicon?
All of them and none of them. This is a complicated topic, but the tests on the testers aren't actual applications - they are snippets of applications. In the creation of the test programs, engineers figure out what the worst applications are and essentially try to write those massive programs as much shorter tests. This is harder than it sounds.

Also using design knowledge - designers know what the worst conditions for their units are - you can write hand-coded assembly that often does better than real applications at testing a unit's function. So the testers don't run real applications - they are running abstractions of real applications.
How long does the test take?
Less than 1 second usually.
Do they test the max clockspeed of each core?
Yes.
For an overview, the first step after fabrication is called sort. It's a wafer test that tests the full wafer. It's done at speed to some extent but power-delivery is an issue so it's not a full power test. Checks are done on the cache, and as many transistors as possible are tested to make sure that they can toggle on/off. Then the parts are packaged and they go to "class" testing which is done at multiple temperatures and voltages and they are binned and fused. Then they go to burn-in testing which is an accelerated stress test environment designed to weed out early failures. Then back to class for post burn-in binning and then they are shipped.
There's a pretty graphical overview of this here:
http://download.intel.com/pressroom/kits/chipmaking/Making_of_a_Chip.pdf (Adobe PDF)
See pages 10, 11 and 12. They skip burn-in but otherwise it's a good high level overview... with cool pictures.
Patrick Mahoney
Senior Design Engineer
Intel Corp.
* not a spokesperson for Intel Corp. *