And how long would it take to read them all out loud?
Randall Munroe answers these questions today at xkcd's what if? page — the answer involves Claude Shannon, a rock 100 miles wide and 100 miles high, and a very long-lived bird (or perhaps a reliable species of birds). You should definitely read the whole thing.
Randall calculates the number of tweets based on Claude Shannon's estimate of 1.1 bits per letter in English text, giving him an estimate of 2140*1.1 ≈ 2*1046 distinct tweets.
Let's check the rest of his arithmetic, just to be sure.
100 miles is 16,093,440 centimeters, and a cube of rock 16,093,440 centimeters on a side is about 4.17e+21 cubic centimeters. We're told that
[E]very thousand years, the bird arrives and scrapes off a few invisible specks of dust from the top of the hundred-mile mountain with its beak. When the mountain is worn flat to the ground, that’s the first day of eternity. The mountain reappears and the cycle starts again for another eternal day. 365 eternal days—each one 1032 years long—makes an eternal year. [...] Reading all the tweets takes you ten thousand eternal years.
So it takes 1032/1000 = 1029 bird-visits to carry off 4.17*1021 cc of rock, or 4.17*1021/1029 = 4.17e-08 cc per bird-visit. That's about 40 billionths of a cc of rock per visit; since the density of rock is about 2.5 grams/cc, the bird has to remove and carry off about 100 nanograms of rock per visit.
That counts as "a few invisible specks of dust from the top of the hundred-mile mountain", or close enough for poetry.
The rest of the calculation seems to come up a bit short, if I haven't made a mistake in arithmetic, since ten thousand eternal years is 10000*365*10^32 = 3.65e+38 regular years, which in turn is about 3.65e+38*365*24*60*60 ≈ 1.15e+46 seconds, rather than the 1047 that Randall calculated that we actually need. But at this point, a factor of ten hardly matters.