I don't think it would flat out prevent it, but it might make it run rough. I thought this happened to me on my gen 3 once (well, a bad sensor at least). I have replaced the sensor once because it started failing and reading 25 psi with the key on and engine not running. Once day my mom borrowed my car and it died at an intersection and wouldn't restart. I thought maybe the sensor went again and it needed to see pressure to kick on the fuel pump or something, but turns out it was just out of gas and the fuel level sensor was bad so it still looked like it had 1/4 tank.
But generally speaking, low/no pressure would mean lifters might not pump up, so valves don't open as far. At least on a brand new fresh engine build I've see the first start sometimes be a little harder, but usually it's more just noisy than flat out non-functional.