The main downside is you can starve pressure-critical items like main bearing journals from adequate pressure to keep metal from touching metal when the bearing load is at the maximum (in the case of plain bearings - roller and ball bearings are lubed by spray and not sensitive to pressure as long as the oil is present). If the journal surface touches the soft babbit bearing material, material transfer occurs and the breakdown process starts. Maybe 50 or 100 hours later, the breakdown is complete and it throws a rod.
Lubricating qualities are a factor, but the main issue is oil flow and the effects of viscosity, being higher at low temperatures, on flow, which affects local pressures. You are pumping oil through a network of passages and galleries that get progressively narrower as oil is branched off to feed different parts. If viscosity is excessive, oil tends to "back up" more at restriction points, and pressure on the upstream side is too high, and pressure on the downstream side is too low. You could say that the oil distribution network is "tuned" to function with a specific pressure range and viscosity.
If viscosity is out of limits on the high side, you end of with different internal parts of the oil network where pressure is higher than the gauge indication and other parts where it is significantly lower (more than the normal variability). If bad enough, some components may get starved of oil, and other areas subjected to too much pressure may start to leak, or even, in the case of oil lines and coolers, rupture. Plain main bearings, that have to directly transfer the force produced by the combustion stroke to torque without touching, are most critical part of this, although everything is affected to some degree.
The temperature of the engine itself isn't that important, except for perhaps cylinder heads. Lycoming's operating policy is, if the oil is in the green and the engine can take full throttle without stumbling, it's warm enough.
Also, if you are using multigrade oil like Aeroshell 15W50, you have a lot more latitude and going to a high power setting while below the green on temperature, on an engine designed to use straight weight winter oil like a 60 weight (equivalent to SAE 30), is much less likely to damage anything.