I do not know this for Hungary specifically, but in many Western countries the cause of death is considered a private matter that is not normally mentioned in obituaries. It is at most hinted at in vague terms such as "suddenly and unexpected" or "after a lengthy and severe illness". Unsurprisingly then, I have been unable to find relevant information in any of the obituaries published after Rényi's passing, other than that he had not been well for some time, with this information being provided by himself:
Leopold Schmetterer, "Alfréd Rényi, in memoriam." In Proceedings of the Sixth Berkeley Symposium on Mathematical Statistics and Probability, Volume 2: Probability Theory. University of California Press, 1972, pp. xxv-l: (online):
On January 5, 1970, I received a letter from Alfréd Rényi in which he wrote of plans to visit Vienna again, and he incidentally added that he was not feeling well and had trouble with his glands. On February 2nd, Professor M. Csörgő of McGill University, who was then visiting at the Mathematic Institute of the University of Vienna, relayed the news that the Hungarian radio had announced that Alfréd Rényi had died the day before.
In Rényi's book "A Diary on Information Theory" (translation of "Napló az információelméletről", Budapest: Akadémiai Kiadó 1982) posthumously completed by his student Gyula Katona based on the author's 1969 notes, he writes from the perspective of an anonymous student of his. On page 55, the imaginary student records in passing:
I am eagerly awaiting the future lectures. The professor doesn't look too well.
In a blog post "The last paper of Catherine Rényi and Alfréd Rényi: Counting k-Trees", dated May 1, 2019, Gil Kalai, Professor Emeritus of Mathematics at the Hebrew University of Jerusalem, states:
A few months after the conference, on February 1, 1970 Alfréd Rényi died of a violent illness.
I have found one source that gives a very specific cause of death. However, no reference is cited for this, nor have I been able to corroborate this information from other sources at this point.
Norman L. Johnson, Samuel Kotz (eds.), "Leading Personalities in Statistical Sciences: From the Seventeenth Century to the Present", Wiley 2011, p. 206 (Google snippet):
Following the unexpected death of Catherine Rényi in August 1969, physicians diagnosed an inoperable lung carcinoma in Alfred; he died on February 1, 1970.