If resistors in your product are failing in the field, there are a few things I'd look at:
Are you buying parts from a reputable vendor? If you're buying from E-bay or a 2nd-tier distributor, you don't know how the parts have been stored and handled, and what quality controls were used when manufacturing them, or even if they are correctly labelled.
Is your design using the parts within their specs? If you're buying from a reputable vendor, an overstress due to a design error is the most likely cause of field failures. Is the operating temperature (including self-heating) within spec? Is the operating current in spec at all times? Are you de-rating the parts appropriately for the lifetime you need?
Is your manufacturing facility handling the parts correctly and following recommended thermal profiles for these parts? Are they following requirements for moisture sensitivity?
Are your customers using your system within specs in the field? Are they using it under the temperature and vibration conditions you expected? Are they using the product at higher altitude than you expected?
If you don't find any issues by answering these questions, then is the time to start looking at testing. (of course, if you've had customer-visible failures, your customers may insist on further testing after you take corrective actions for whatever issues you found through design analysis)
For a problem like this, HALT/HAST testing ("highly accelerated life testing" and "highly accelerated stress testing") could be appropriate. This involves operating your system in extremely aggressive environments, with high temperature, humidity, and ongoing vibration. The environmental stress is increased until the product fails. The failure mode is analyzed and corrected, and then stress is increased until the next failure mode is observed, and so on. The idea is to find the weakest areas of the design and improve them, iteratively, until some fundamental limitation is found or increased stress becomes impractical.
But HALT/HAST is a system-level test, and you asked for a component-level test. For component level, accelerated life testing is also possible. Generally, based on some physical knowledge of the expected failure modes, you can predict (or assume) an acceleration factor for operating the part at high temperature. For example you might find that some type of failure doubles its occurrence (halves its MTBF) for each 10 degrees of increased temperature. In that case, you could simulate 10 years of operation in 3 months by operating at 55 degrees above the design limit.
However, good estimates of the acceleration factor depend on good knowledge of the important failure mechanisms and are best if validated by long experience with the particular types of parts; so its best to get this information from your vendor. Also, there's a limit to how much acceleration factor can be used --- at some point you could accelerate some secondary failure mechanism so that the failures you see in the test don't represent the type of failures you see in the field.
For PC's made after the introduction of the ATX, the power switch is a logical input to the motherboard--the same kind of input you want to apply to the Teensy. The motherboard, in turn, outputs a separate logic signal to turn on the power supply.
In my experience, the reset switch is as good as standard, as it dates back to the original IBM PC (which didn't have the switch, but did have a connection on the motherboard for one). This is not so for the power switch. I have seen schemes where it pulls a signal up and neither side is grounded. Since you have found one side of your switch is connected to ground, that makes your connection easy. Simply attach a voltmeter to the two pins, and observe the voltage present there. Verify that this voltage drops to zero when pressing the switch. Your Teensy will see this on its input. It doesn't matter how they implemented this. If it's a floating input they will have a pull up resistor already there to make it work. Your input is just watching the resultant voltage level.
Note that the power supply is turned on when the motherboard drops the #PS_ON signal to ground. This is the infamous "green wire" on the ATX connector. If you bring this signal into a second input on your Teensy, you will now be able to tell if the button was pushed while the computer is already running (because it will show a logic 0).
Also note that certain voltages are always present even when the computer is off (or in standby). This is how the power switch manages to have power when there's no power. This is also a heads up that you may find power in other unplanned places (like some USB connections).
Best Answer
These a filtering caps on one of many of platform voltage rails. Capacitors blow up only due to overvoltage, or too much of ripples/ringing. Which means that some active components (power MOSFETs) are likely blown as well, causing this, and likely some other damage. These caps can be high-value (47uF-100uF) low-voltage (2.5V-4V) caps. Without proper schematics for this board it is not possible to repair it. And the schematics will be impossible to obtain. So throw the board away, or keep it for re-use its components for DIY projects.