Both UV and IR are harmful in enough quantity. Actually visible light can be harmful at enough intensity too, but it is immediately painful, while UV and IR are not.
The amount of UV and IR emitted (especially UV) are dependent on the temperature, and since oxy-acetylene is still hotter than the surface of the sun, it produces more UV than staring into the sun, and plenty of IR to be dangerous.
IR is a radiated form of heat. It's [most of] the heat you feel when your hand is near a glowing weld, or the warmth on your skin from the sun.
Since the retina has no cells that sense heat (unlike your skin), sufficient IR (which is focused on the retina by the eye's lens and cornea), will burn up the retina like an ant under a magnifying glass in the sun, WITHOUT ANY PAIN. Just to restate that, your retina is no more capable of sensing heat, than your tongue is of tasting "blue", so it is easy to cause permanent burning of the retina, with no warning such as pain. Fortunately, low IR exposure isn't thought to have long-term effects (that is until you pass that threshold of burning).
UV is typically associated with long-term eye damage (cataracts and slow irreversible damage to the retina, etc.), however I believe that it is also the cause of arc-flash burn (which is very much like a sunburn on the cornea), but your ophthalmologist is better qualified to clarify that. Thankfully, most plastic lenses today (even clear glasses, and especially sunglasses) block a large proportion of UV (like 99% or so).
While 1% of the UV emitted from an electric arc welding process is still VERY dangerous (since the significantly hotter electric arc emits MUCH more UV), that remainder from oxy-acetylene should be ok.
Our bodies have have no way of sensing UV, or immediately sensing it damage. On hazy days, we're more likely to be sunburned, because the IR (which we feel, and which makes our skin feel hot) is more blocked by the haze than the UV, and so we allow ourselves to be more exposed to UV, since we don't feel the damage.
By this point, I'm expecting someone to respond:
after they fell asleep on their keyboard, but if you're still awake I just want to be clear that I can't put this one to rest.
There probably isn't enough IR in OA to cause retinal burning, but if it does happen, it will happen without warning, and the damage is permanent.
I can say that mirrored sunglasses will block MUCH more IR than cheap chemically tinted glasses, so if you must use sunglasses, mirroring should offer decent IR protection for OA.
There are many kinds of dark tints out there, that block a lot of visible light, but are almost transparent to IR, so be careful. Lenses with a shade number are guaranteed to be using a tint that blocks IR and UV as well as visible, across the board.
Just about any plastic glasses should afford decent UV protection for OA. My optician actually has a spectrophotometer, which can measure exactly how well my glasses block IR and UV. That may be something you might want to ask about.
Still, I ALWAYS use goggles (or my auto-dark welding mask, turned off), for cutting, because of the danger of a spark bouncing in my eye. I've already had a steel shaving removed from my eye, and never want to go through that again.