How long will a CPU last running 24/7

CPU failures from ordinary use are very rare. Manufacturers typically provide the default "it isn't going to fail" figure of 100,000 hours, which is just over ten years. But most likely, it will continue to work until it's technologically obsolete.

A CPU itself will probably last tens of years.

The problem with system reliability is actually all the other components in the system. The CPU will undergo thermal heating and cooling cycles as the load changes and as you turn the system on and off. The same is true for pretty much every component in the machine. The CPU silicon die is connected to pads or pins on the CPU by tiny bits of wire which will be affected.

Hard drives will spin up and spin down, resulting in mechanical stress on the motors and thermal stress as the drive power controller warms up and cools down.

Damage to the silicon in the processor is unlikely to be any kind of failure mode you will see, the voltages are low and the paths are well designed. It is only in NAND Flash memory devices that you are likely to see a failure due to silicon insulation breakdown, but then that is because they are intentionally driven to cause a non-catastrophic breakdown (but it eventually will be catastrophic).

You are more likely to see problems due to thermal or mechanical stress, as temperatures change and components move very slightly, than you are to see any failure within the actual CPU silicon.

Thermal failure is the only mode I've seen personally for a CPU (as opposed to the support equipment). The last time I saw (or heard) of this was with an old Pentium III laptop that went on fire (literally) one morning.

Thermal management is now a much more sophisticated operation in systems and so CPUs are not allowed to run too hot. Typically a CPU will throttle it's clock to reduce temperature if need be, and the whole system can shut down if it detects a temperature that's too high.

So I think with modern CPUs failure (before obsolescence) not a real issue.

Motherboard failures seem more common (but still comparatively infrequent) than CPU failures. Drive failure is common as dirt, particularly in heavy duty server settings. I'd not give a thought to the CPU any more.