I'd like to say first that I'm not an historian, but a Pol Sci researcher so my take on the 1840 election is from a political communication point of view.

From all the information that I gathered on this very peculiar election, Harrison' health wasn't a significant issue during the campaign. His age could have been, and the Democrats kinda tried to make it one when one of them said that Harrison would be better off staying in a log cabin, thinking about life and such, like a retirement thing. Note that on this point, the focus wasn't specially on Harrison age but it could have been interpreted that way. Anyway, it was a communication failure since the Whigs span the whole "log cabin" thing into a way to make Harrison "a man of the people", using imagery and story-telling to create a likable character from scratch.

So to answer your question, as far as I know, I don't recall any instance of big debates and discussion about Harrison health and/or age. But then again, I only focus on the campaign communication aspect and am at the beginning of this research of mine, so maybe some historians could go more in depth on the topic.

Sources : Carwardine, R. (1983). Evangelicals, Whigs and the Election of William Henry Harrison. Journal of American Studies, 17(1), 47‑75.

Chambers, W. N. (2012). Election of 1840. In History of American Presidential Elections, 1789 - 2008 (Vol. 1). Facts on File, Inc.

Gunderson, R. G. (1977). The Log-Cabin Campaign. Westport, Conn: Praeger.

Heale, P. M. J. (1982). The Presidential Quest: Candidates and Images in American Political Culture, 1787-1852 (1st ed). London ; New York: Addison-Wesley Educational Publishers Inc

Just a disclaimer, but due to Harrison's limited time in office and lack of much real successes there hasn't been a lot written about him in the last 100 years.

Harrison's age or health was never a serious issue in the campaign, though not for a lack of trying. His Democratic opponents tried to make it into a problem but it never stuck. They called him names like "Granny Harrison" and attempted to make him look like an old senile out of touch farmer who has little knowledge of the goings on of the world.

Unfortunately for the Democrats, Harrison's camp twisted this into a positive thing for Harrison. They recast him as a simple farmer who represented average folks of America, who had been toughened by war and the west. They also portrayed Van Buren as a fancy east coast elite. The strategy worked and Harrison won.

So while his age was an issue, it was never a significant problem for his campaign. I don't see much evidence that people seriously worried about his age or health during the campaign, it was mostly just political attacks. While nearly all Presidents were younger than Harrison at their inauguration, even in the early 19th century well to do people could live quite a long time. Besides Washington every President had lived into their 70s, and most to around 80.

Further to this, has anyone ever died before reaching office? Either as a presidential candidate or President Elect before inauguration day? What action was taken in this case?