All on the tin, really. What if the blowhard had bundled up or cut his speech short and never caught the fatal pneumonia?
A substantial boost to the strength of the Whig Party. Tyler was really a Democrat, and soon broke with Clay and Webster, vetoed Whig legislation, and didn't give patronage jobs out to Whig operatives.
If Harrison is in office instead, the Whig program goes right through, and Federal patronage goes to build up Whig organizations.
The first impact is in the elections of 1842-1843. I don't think the disastrous showing of the Whigs can be entirely reversed (the Whigs lost 67 of 139 seats), but they probably do somewhat better, holding 85-90 seats.
The second impact is over the annexation of Texas. Harrison will not push for it, as Tyler did.
The third impact is the 1844 election, which Clay nearly won. With the support of Harrison's Whig administration, I think Clay pulls it out.
There will be no Mexican War until after Clay leaves office in 1849, if then.
The biggest impact would be the actual implementation of the Whig program. For instance, the creation of a Third Bank of the U.S.
Also the program of "internal improvements". One obvious place for Federal action was to clean up the Mississippi - removing snags, marking shoals, and setting up navigation lights. It wasn't done, AIUI, because of the Democrats' opposition to "internal improvements".
Once such a program was in place, with obvious benefits, the precedent would be tremendous (as would the political benefits to the Whigs).
Is it possible the Whigs could become a long-term dominant party?
The Supreme Court would be affected, of course. Tyler appointed only one Justice, Samuel Nelson. Tyler nominated several Democrats; all were rejected by the Whig-controlled Senate until Tyler nominated the cautious moderate Nelson.
Polk named Democrats Levi Woodbury and Robert Grier. Clay would of course name Whigs.
Yet another impact: if there is no Mexican War, the U.S. does not annex Utah. The Mormons get to build their independent state of Deseret for several years, unless Mexico tries to march against them (rather unlikely, IMHO).
If there is no Mexican War, there would be no Crisis of 1850 over the status of the Mexican Cession lands.
But there would be a different political crisis with the sudden influx of a huge number of Irish immigrants (and Germans). The Whigs OTL made common cause with the Know-Nothing "American" Party; this was mostly a tactical maneuver by the party remnants; but ISTM that the more upper-class Whigs were naturally inclined to react badly to Catholic Irish mass immigration.
This might lead to restrictions on immigration or naturalization being enacted - not as drastic as OTL's Know-Nothing's wanted, but still more than happened until the 1920s.
On the subject of immigration: no Mexican War, very probably no Gold Rush to California, and no Chinese immigration there? Perhaps settlement increases gradually until there is Texas-style secession from Mexico?