Wood is full of air, and air is a terrible conductor of heat. It's not as complicated as it sounds, lighter, i.e. less dense woods, translate heat more poorly than dense ones.
If you look at a cross-section of a piece of wood on the microscopic level, you'll actually see that it's a network of relatively free-floating tubes within a strata of connective resins and polymers, which eventually dry out and allow air to penetrate once removed from the tree. Those tubes are used by the trees to carry things such as nutrients and liquids throughout the plant's various types of stalks, and they are also used to provide structural support. The direction the tubes are going in is the wood's "grain." heat travels down the grain relatively easily, as the tubes are solid pieces from start to end, whereas heat cannot travel very well transversely across the tubes due to the air within and around these tubes being absolutely terrible at conducting heat.
Think of it similarly to the protective ceramic plates used to protect spacecraft upon reentry to the earth's atmosphere. These tiles can reach temperatures of over 2000C, but can be held by an unprotected hand at the same time due to how poorly that heat is conducted through the surface. Skin has water on it, and within it, and water has a very high specific heat, which is a measurement of how many Joules of energy is required in order to heat one gram of material by one degree in the Celsius or Kelvin scales. So our skin has a very high specific heat, meaning it can absorb large quantities of energy while remaining at a fairly constant temperature. Since heat propagates very poorly through materials like the ceramic in question and wood, it's a very simple idea.
There is simply not enough energy being transferred to your skin quickly enough for it to harm you. The medium is incapable of transferring the provided amounts of heat in such a way that it will cause you harm, as the heat that is absorbed by your skin is not replaced by heat residing in other places within the medium due to its incredibly poor conductivity. So, once your skin makes a "cool" spot due to contact, that spot will stay cool, especially considering the fact that water is much more conductive of heat than those other materials, meaning the heat dissipates through your tissues and warms your body rather than burning a single localized spot.
In regards to your query about the wood being exposed for a particularly long time to the same temperature, it is much the same as an object reaching terminal velocity. It is impossible for the object to change when the system it is within does not change. The hotter an object is, the more quickly it will radiate the heat it stores, since "Nature abhors a vacuum." It will eventually reach equilibrium within its system no matter what, so long as the system remains unchanged. If you were to turn the sauna hotter, or cool it down, the temperature of the wood would change gradually, but it will always reach an equilibrium at which point the energy flowing into and out of the wood in the form of heat do not surpass each other.