I don't know if any explicit influence came from deism, but I think the detached nature of the Valar is quite reminiscent of Deism (and aspects of Roman Catholicism, for that matter.)

Deism relies heavily on the concept of a Mechanistic Universe, seeing God as a great clock-maker who created a perfect machine that runs on its own and does not need interference; to do so would damage the cogs and springs that keep it running harmoniously. Many of the founding fathers of the United States (Benjamin Franklin and Thomas Jefferson in particular) were staunch deists, which explains a lot about their peculiar paradoxes (owning slaves, taking mistresses, acknowledging their illigitimate children).

This naturalistic approach seems very similar to the "religion" of the elves, if there is such a thing. Recall that while in Lothlórien, the Hobbits were greatly impressed by the "magic" of the elves, who became confused at the use of such a word to describe their skills. For the elves, these spiritual and physical gifts were more inherent than that. Their connectedness with nature and their ability to sense and be influenced by the world around explains their lack of need for organized religion. To them, "religion" would be merely a mannish concept for those who are unable to seamlessly incorporate spirituality into everything they do.