In the title of the poem 'Do not go gentle into that good night', is 'gentle' an adjective referring to the poet's father or an adverb referring...

Your second suggestion is correct; the word "gentle" is used here as an adverb. Thought normally one would use the word "gently," Thomas is using the word in a poetic context. But it begs the question, why did he use the word "gentle" and not "gently" as one would expect? I have loved this poem for a long time and always appreciated the use of the word "gentle" and was never bothered by its problematic grammar, if indeed there is even a problem!

But I did some research online and found a fascinating discussion where someone asks if this indeed grammatically correct. The response is intelligent and thorough and also offers information about the poem. In part, the responder says:

"At first glance, Thomas could have chosen to phrase his exhortation “Do not go gently into that good night,” and it would apparently have meant much the same as “Do not go gentle into that good night”. 

However, the situation that gave rise to the poem indicates to me that Thomas preferred ‘gentle’ to ‘gently’ because he wanted to underscore his desire for his father’s lack of gentleness to be an existential state in the face of approaching death – perhaps with the underlying hope that by fighting against oncoming death, his father might actually prolong his life; his father was not actually on his deathbed at the time the poem was composed. 

‘Gentle’ as it is used here is therefore a synonym for ‘meek’ or ‘submissive’."

I like this very much: the clarification that "gentle" is a synonym for "meek" or "submissive" as it emphasizes Thomas' wishes for his father to prepare himself for death, and to not give into it (or the pain or indignity that might come with it), and to fight to stay alive. It is often said that one's emotional and psychological state have a great deal to do with one's ability to stay heathy. Of course, there are exceptions, as when severe illness or accidents cause a sudden death. But Thomas is here contemplating not just his father's attitude towards aging and death, but very likely his own. He is recommending approaching death with the same passion and optimism with which one approaches life.