Dave Smith of Retrosheet tried to address this question in his presentation at SABR 36 in Seattle.
You can read the pdf version of his presentation here.
From 1957 through 2005, teams score the most runs per inning when the #1 hitter leads off, .55 runs per inning. Teams score the fewest runs when the #7 hitter leads off, .40 runs per inning.
The first inning is the run-scoringest inning of all. (That's a new word created by me.)
I found it interesting that in Dave's study, that there has been one game in the majors since 1957 where the leadoff hitter in the batting order did not bat leadoff. That was in this game in 1961 when the Angels #8 hitter, Gene Leek, thought he was supposed to bat first and lead off the game. Since he made an out, the Red Sox didn't care. The Angels figured it out by the second inning.