I think there's strong evidence that lung trauma caused by airspeed is not an issue, but we can analyze it. Sorry about the length. I treated this sort of like one would treat it on Physics.SE
Human lung alveolae can withstand 60 to 70 mmHg of transpulminary pressure before rupturing. The worst case for the scenario you describe would be at the diastolic blood pressure. From this study, I draw the conclusion that 80mmHg diastolic blood pressure is not unreasonable (that was measurement during exercise, and I am certain one's body is under duress while flying a hundreds of miles an hour unprotected). This means trauma can occur at 140 to 150mmHg pressure. As the air in the trachea and lungs would be relatively stagnant (unmoving), I'd conclude that a stagnation pressure of 140 to 150mmHg would be required for trauma to the lungs. We'll convert to SI and round both of those measurements to somewhere in the middle: 19kPa
From Bernoulli's equation, we can calculate the required velocity of air if it stagnated. P1 = P2 + 1/2*rho*v^2, where rho is the density of the fluid (air), thus ΔP = 1/2*rho*v^2. Rearranging this we get v = sqrt(2*ΔP/rho). Your question does not specify an altitude, but since you are talking about motorcycles, let's just assume density at sea level: 1.225 kg/m^3. This gets us a velocity of 174 m/s.
This is an incredibly conservative metric. First off, we're calculating to trauma, not to death. Second, this assumes the air stagnates. If the nostrils of our unfortunate high-speed individual were tied to a pitot tube (a device designed to measure airspeed by stagnating the air) pointed into the wind, this number would be accurate. In practice, the nostrils will not form such an ideal structure, and the air moving past them will remain at a high velocity, dramatically decreasing the effect seen by the lungs. A human subjected to these things will be in pain, and likely to strive to avoid such angles.
The third issue here is that we're assuming they're breathing. If we're interested in long term effects, they're probably going to have to keep breathing or they will pass out. In the short term, I'd be quick to clamp my glottis shut to protect my lungs as an instinctive reaction. I did not find any literature on the maximum pressure of human glottis, but this article did mention that canine glottis pressures reached 53kPa during barks, so the breaking point should be substantially higher than that. Adding this in would push the necessary velocity up to 330m/s, stagnating. This is roughly the speed of sound. And since this is a stagnation velocity, unless your poor subject crammed a pitot tube up their nose and faced forward, the actual speed where this damage would occur would likely be much higher.
I can't find the study, but NASA did a study which showed that the human body would be ripped apart by supersonic winds at sea level. Those who have survived at these speeds have been at altitude. We can consider the case of Bill Weaver, SR-71 pilot who had his plane disintegrate around him at mach 3.18. He was flying at 78,000 feet at the time. Up there, air density is a mere 0.06391 kg/m^3. Plugging the numbers into the same equation yields 771 m/s. At those altitudes, mach 3.18 was around 940m/s, which isn't much more than this stagnation pressure. So even if Bill Weaver lacked his safety gear, his nose would have to act as a pretty effective pitot tube before lung trauma started playing a part.
... and at that point, the lack of a parachute would be more dire.
So in conclusion, if you're flying a SR-71 naked, and the plane disintegrates around you, do not intentionally orient yourself feet first while tilting your head down to get a better look forward. Instead, orient your head in absolutely any other orientation you please... and think about your life choices.
Also one could conclude that one should not put a pilot tube up ones nostrils before skydiving out of a supersonic aircraft, but from a medical science perspective, we can probably generalize that a bit and say "don't put aircraft parts up your nose.". Seems like solid advice, regardless of skydiving plans.