Normally no. Rule of thumb is: if it's an '030 or older, or has soft power, it probably doesn't need one. Now this isn't a hard-and-fast rule: there are several machines, notably the 6-slot Mac IIs, that won't even power on without good batteries, but generally the older models don't specifically need a battery to boot. This changed with the '040 and later models with hard power in that they wouldn't boot properly without the battery. The LCs are kind of in an odd bracket in that they can be fussy without a PRAM battery but normally that's because they're not used often enough and/or some capacitors are going bad; normally they don't care that much about the PRAM battery.
Basically the power manager has an internal 8-bit CPU that manages power-on functions and the PRAM battery keeps its parameters loaded and ready to go as soon as it gets main power. If the battery in hard power machines dies or is missing, the CPU loses its data and its operating state so that when power is applied, the power manager's CPU has to boot first before it can perform any useful work. However, by the time it's booted, the power-on initialization window for the Mac has passed, so the machine just sits there; you need to mash reset or do a quick off/on flick of the power button to "jump start" the computer. This procedure cuts the power just long enough to force the other systems in the Mac to go back to their initial power-on state but not long enough to drain the power manager's CPU, so it stays loaded and can properly initialize the system.
Why do soft power machines work without a PRAM battery? Because as soon as they're plugged in, the power manager starts receiving +5v from the power supply, which allows it enough time for it to boot and watch for power-on signals.
Why do weak PRAM batteries cause problems? Because they don't put out enough voltage to maintain the power manager's CPU, so it becomes corrupt and/or crashes. The only cure for that is to hold down the PMU reset button for a while and/or remove the battery and unplug the computer for a bit until the power drains away completely, allowing the power manager's CPU to reset and load with a fresh slate the next time.