How can I modify the programmable interval timer value stored in the computer so that the system clock runs faster?

According to this Wikipedia page (https://en.wikipedia.org/wiki/System_time#Implementation) computers use a programmable interval timer (PIT) that periodically interrupts (when reaching a certain value) the CPU with a timer interrupt routine, adding one tick to the system clock.

Uff. That's a bit of a broad statement, but let's assume your CPU and your operating systems do that. OK.

Could I make the computer go faster if I alter the value required for the PIT so that it is lower?

No. That has nothing to do with the speed of your computer.´

that way the PIT would generate an interrupt more frequently and the system time would increase faster.

Highly undesirable. That means the actual work that needs to be done is interrupted by a routine that actually contributes nothing.

My objective is to make the computer execute time-sensitive processes faster than it should.

Then you'd program a timer interrupt for your problem, instead of depending on the system tick!

Linux has the High Resolution Timers API, and I'm sure Windows and other operating systems have similar functionality.

For example, when playing an audio file, the audio would sound accelerated, or when using an instruction like time.sleep(5) in python, the real-time would actually be lower than 5 seconds. Is this possible?

Neither audio handling, nor sleeping, require any better precision than the systick would have anyways, AND audio handling might very well be interrupt driven by the actual audio hardware, anyway.

So, you're really barking up the wrong tree here. I think your computer is much faster than you think.

Modifying a default timer is not the solution (aside from breaking thousand other things). If you actually needed low-latency processing, again, your program would directly interface with timers itself, or with interrupt-generating hardware with no or a very thin layer of hardware abstraction through your operating system.