I have been rather unsure about this for a while. It clearly boosts the healing you do to yourself from abilities and Spellvamp/Lifesteal, but does it affect things like the Force of Natures percent of health a second? Or how about just the flat regeneration from items?
I would say yes to all of those questions if I'm understanding the discussion on the wiki about it properly.
As far as I'm understanding it Spirit Visage seem to affect all healing sources of every kind (spell vamp, regeneration, Lifesteal, ect) EXCEPT those that come from outside sources. So - If your healing in any way from something YOU did, yes it applies, but not from something a teammate has done.
While this would include flat regeneration and Force of Natures H\ps, keep in mind that some sources of healing (regeneration in particular) are so small that they only matter over time anyway, and as such, the boost they would get from Spirit Visage is so small it would be nigh unnoticeable.
http://leagueoflegends.wikia.com/wiki/Spirit_Visage has just about all the information you could want about it.