Well the ring was essentially the essence of Sauron,so it probably worked against Iluvatar's influence,making him rely on his agents like Gandalf,who ensured that it was passed on to Frodo.The greatest risk with something like the One Ring is that is corrupts whoever tries to wear it,in many ways Iluvatar made Frodo's and Bibbos lives a misery just to make sure that no one else (like say,one of the Maia) could become corrupted and cause an even greater discord.Saruman is corrupted by Sauron anyway and it's difficult to say wheather Iluvatar sacrificed him because he knew he would fail,or because did not forsee another of the Maia being corrupted by sources like the palanti.I think Iluvatar knew he could afford to lose Saruman,but not Gandalf,without another agent to help guide the ring bearers his plan to destroy the ring would fail.
Long story short,if ttoo many Maia become corrupted its game over,but he needs one agent standing in order to win.
It's difficult to say who was who Iluvatar could influence,certainly there are many forces which are working against what he wishes(like Sauron and the ring itself) and those he most likely has no influence over like Tom bombadil,either because Tom too important to risk or because Tom doesn't pay too much attention to Iluvatar and he's prepared to leave Tom alone,but that does't mean he's not watching him closely through Gandalf.
