I read on wikipedia that the reason women gave birth in saunas was because of the sterile, warm environment with water readily available. But there must be more than just sterility and warmth that made this such a common practice in the Finnish culture.
The article goes on to state:
In Finland and Estonia, the sauna is an ancient custom. It used to be a holy place, a place where women gave birth, and where the bodies of the dead were washed.
Additionally, the sauna provides the following benefits that lend directly to easing the effort of childbirth:
The temperature changes of therapeutic sauna can help and this has other benefits as well. When first used gradual increases in heating and cooling are recommended. Therapeutic sauna reduces stress hormones and the cardiac workload is considered about half that of a walk, so initial exposure time is important also. The hypothalamus in our brain controls the balance homeostasis of the autonomic nervous system between the ACTION sympathetic and the RELAXATION parasympathetic nervous tone. The well known ‘fight or flight’ stress response produces hormones intended to be burnt off by action, but in a modern lifestyle such hormones may remain in the system. Chronic illness can be associated with altered sympathetic nervous function. Continual stress may alter the balance point of homeostasis, as can some persistent viruses. Allostatic load measurement is an emerging science of measuring with physiological tests the accumulated effect of all types of stress, over time, on the body. Four different patterns of dysfunctional allostasis have been identified, each associated with certain chronic conditions. When allostasis (the process of maintenance of homeostasis, adaptation, and survival) is dysfunctional the balance point is shifted and persistent symptoms may result. In one form of dysfunction the hypothalamus and HPA axis responsible for producing hormones is found to be hypo functioning with effects on the sympathetic system and the immune system. In particular production of hypothalamus controlled HPA axis hormones such as ACTH and cortisol; as well other hormones are affected. Other patterns of dysfunctional allostasis involve conditions where there is failure to habituate or adapt to stress and another pattern with high levels of stress hormones, causes conditions such as hypertension or high blood pressure. Therapeutic sauna has been shown to aid adaptation, reduce stress hormones, lower blood pressure and improve cardiovascular conditions.
For more information about the effects of saunas as a health practice, see the above cited link. I recommend the article there as it is heavily cited with biomedical research. Unfortunately, I am unable to located any research directly connecting pain relief and comfort during birth to the use of saunas.