Are interrupts needed for a computer system to work? Could you have a computer system (hardware and software, including the OS) that worked without an interrupt mechanism?
No, they're not needed for a computer to work. For instance, you can use polling to interact with devices, instead of interrupts, and you can use cooperative multitasking for scheduling multiple processes, rather than preemptive multitasking.
(Thank you, @Paul A. Clayton!)