I've given "'His Accidency' John Tyler" by John Wicker in
What Ifs? of American History a re-read after seeing this thread. A typical essay for these Robert Cowley edited collections featuring little examination of how things might have changed or what might have happened instead.
I think there's three options for John Tyler not just taking the role of presidency, since there wasn't even any debate to him since he arranged his own inauguration as soon as he cold get back to D.C. following Harrison's death: the first is Harrison simply doesn't die, and the precedent is left to be set later; the second as
@napoleon IV has suggested sees a different VP for Harrison, one that isn't quite keen to start redecorating as soon as his predecessor is planted; the third, suggested by
@zaffre, still sees Harrison die with Tyler as his VP but a year or so down the line.
As to the first, well there's a lot more to consider from a surviving Harrison than what's Johnnie T up to, perhaps he becomes Calhoun to Harrison's Jackson. He might just occupy the position and do nothing, wouldn't be the first or the last time; or he might just alienate the President, Cabinet and Congress criticising the American System and calling for a more bullish stance on annexing Texas. Would be amusing, however unlikely, for someone to start Impeachment proceedings against the Vice President.
For the second, there seem to have been three other potential VP picks - all Southerners since Harrison was seen as a Northerner, also all seen as Henry Clay supporters since Clay wasn't too happy about not receiving the nod. John J. Crittenden, supporter of Clay and fellow Kentuckian, was the first to be named and really was a peace offering, but the Kentucky delegation still seemed to pissed off about Clay not getting the tip for top job and declined. John Bell was then offered, but again the Tennessee delegation declined, though for less clear reasons than the Kentucky delegation did for Crittenden. Willie Person Mangum, who would have been willing but turned it down as he was a member of the Convention and considered it improper, was the last potential candidate before Tyler. I think Bell might be the most plausible option with a bit of cajoling, but Tyler was both a Southerner and, at the time, a supporter of Clay, and accepted. Difficult to imagine a 'Stop Tyler' campaign without hindsight. None of the others has the alliterative appeal of 'Tippecanoe and Tyler Too' either.
The third option offers more possibilities. Say someone has the sense to trim down his speech further, or convince him of the benefits of a hat and coat, or to cut it short when the rain starts; then in the winter of 1842 he is caught outside in a downpour completely hatless and dies in similar fashion to OTL. Tyler will probably behave exactly as he did in 1841, but here there's been a year of the American System, a new Bank of the United States, and cooperation between Executive and Legislative branches. There's also elections to the House of Representatives later in 1842, which might make a special election more palatable. Tyler might start making a lot of noise about annexing Texas, which wouldn't endear him to Congress or Cabinet much more.
Suppose it depends on how well things have been going 1841-2, and how much Tyler can be portrayed as an interloper rather than an internal Whig Party fight. Tyler might also go through with a Quixotic run as an Independent for the Presidency, but can't see him winning it and might actually allow the Whigs (likely running Clay though I wonder if Scott might pull ahead of him) to slip through by drawing votes away from the more bullish on expansion Democrats (possibly running a Van Buren/Polk ticket). Wonder if this will be a special election conclude the present term with another election in 1844, or if this will be the start of a new four year term so the next election will be in 1846. Either way, the next election is likely to be a very divisive one.