Hi Neil,
It all depends on how you size and operate your turbo.
In general, the pressure relief, or pop-off valve is not needed in aircraft. Those come more into play during the fast throttle closures that occur in highly boosted race cars during shifting. They spend almost all of their time closed. These valves are not intended to be the primary pressure regulation and would not function well in such a capacity. They only limit peak pressure but do not do so with a high level of accuracy or control ability. They would also be highly inefficient if used in this manner, as the turbo would be producing continuously higher back pressure than needed. The turbo would be short lived and your fuel consumption would suffer greatly.
The wastegate issue is a much bigger question but the short answer is that yes, you probably want/need one. The only exception might be a large turbine turbo that is sized so big that the max boost was only achieved at WOT operating conditions. Even then, you would be limiting the usefulness of your turbo at altitude because if it were sized for WOT at sea level, it would not be very useful at altitude. If sized for WOT at altitude you could use throttle to control boost down low, but you always run the risk of overboost. Similarly, a fixed exhaust bypass would allow a smaller turbo to be used in the same manner but with less efficiency and shorter life. I have experimented with my turbo in this configuration. By removing the flapper valve from my wastegate and creating a fixed bypass I was able to run at WOT and have approximately the boost I wanted, but only at a very specific altitude. Any higher and boost drops off precepitously. Any lower and throttle is needed to control boost. There is not much of a sweet spot as the natural variations in conditions created a oscillating situation that is simply not tolerable. But for maximum benefit from your turbo, longest life and efficiency there is no substitute for a wastegate of some sort.
Dave Leonard
N4VY, Turbo Rotary RV-6