First up is Albert Pujols. He's certainly the best hitter of the generation, and he's had multiple teammates bat in front of him in the line-up through the years. This data, for reasons I won't get into, excludes the 2011 season. Here are all players who batted in front of The Machine at least 100 times in a season, and their OPS with and without him on deck:
Naturally, the results don't come screaming out in favor of the 'theory'. But a few more batters benefited than did those that did worse, and those that benefited, really benefited. And there was a .054% positive change in OPS on average.
I would hate to build a strawman here, so I want to use caution when assuming what purporters of the theory actually believe, but I will concede that I considered that OPS may not be the best instrument to measure the suggested effect. Assuming that Pujols's protection offers the batter "more pitches to hit", an increase in OPS may actually indicate that the batter is walking more often, driving up his OBP, and therefore the protection effect is absent. So perhaps we should look at SLG and BAbip, to see if the batter is benefiting from those alleged "pitches to hit" or if he's just wasting them.
Generally, BAbip increased, .031 on average. And Walk rate did drop, but only by .09% on average. But there's just nothing convincing.
When I tried to get an even larger sample with an arguably more dangerous hitter, the Protection effect seemed to disappear.
HANK AARON'S PROTECTEES
For Hammerin' Hank's Protectees, the change in each of these 4 metrics, OPS, SLG, BAbip, and BB%, was statistically insignificant or exactly zero.