Say for example, one of the cores in a quad core computer overheated and died, would the system compensate by routing all operations to the remaining 3 cores, or would the computer be inoperable? If the system compensates, could it compensate for 3 of 4 cores failing?
It largely depends on the exact failure (and the system architecture, but I'll assume you're talking about a standard x86 based system).
In short, the system will not function properly - with exact symptoms ranging from appearing to be okay "most of the time" to a complete failure to boot. There is no mechanism to ignore the bad core at run time, and as such anything that attempts to execute there risks failing/corruption/crashing.
A CPU core cannot be compared to a car engine. Depending on the system BIOS, it may or may not detect a core failure. Most probably it will still route operations to the core and fail. The BIOS may allow for discrete core disabling (trial and error on each core), in which case you will be able to operate normally. At the end of the day, time to replace the CPU
I can say that I have a i7 6700k with 3 out of 4 cores working. If I have all 4 cores active, it will just blue screen when I start my computer. However, when I go into BIOS and change to only 3 active cores, it will perform normally. I have no idea how the core was damaged as I bought this computer broken for a cheap price and fixed it.