In a word, no ... All of the documentaries and biographies I have encountered mention nothing of an addiction to drugs, either personally or close relationship. With that said, it is not impossible though.
But, Tolkien is known to be devoutly religious, however, not a fanatical zealot, so the 'addiction' could have been one he recognized in that respect. There is also the thread of anti-industrialization that pervades LOTR. The greed (both money and power) that seems to accompany industrialization could also be seen as addiction, and tie closely to the one (gold) ring and the desire for power of Sauron and other characters - Saruman, Boromir (out of desire for good and tragic), and Denethor.
Tolkien would have also witnessed battlefield medicine and other medical 'advancements'. Morphoine, cocaine (as sweetner) and arsenic (as preservative), opium and meth (as mentioned in comments), but also the introduction of ether, laughing gas, valium, and other pharmaceutical 'remedies'. And then of course there is alcohol.