no the computer should turn on, but after that its questionable. The motherboard has a built in processor, comparable to a cmos, to get everything on and get the processor going. a dead power supply would make it not turn on at all.
also a side note, a power supply and a motherboard is all you need in some cases. It will beep at you signaling problems. Also in some cases I've seen a computer refuse to turn on without a PS/2 keyboard plugged in.