So the hypothetical situation is a hitter that never puts the ball in play, no walks, no HBP or other way on-base, never advances a runner, and always strikes out. How many pitches per AB does this player have to take/foul to be valuable by forcing the starter from the game earlier than average (not exactly sure what they meant by the "valuable" term, but I'm comparing to a replacement hitter) (also, this hypothetical non-hitter could also be valuable by burning through the opponent's entire bullpen, but that's not what I'm considering)?
Assuming avg defense (baserunning is irrelevant) as a SS (to be kind), the hitter just has to earn about 14 runs below avg with the bat to be replacement. The problem, and the part that I'm a little shaky on, is that making 600 outs (based on 600 PAs) would produce approx -107 runs. So, the non-hitter would have to produce around -93 runs over 600 PAs by taking/fouling off pitches.
Based on 2012 (by-team for simplicity),
avg team's starters: 4.09
best team's starters: 3.39
avg team's relief: 3.88 FIP
worst team's relief: 4.54
Taking an average starter and replacing him with an average reliever is actually about even, so there's no value to be gained, possibly even generating negative value, by forcing the starter from the game early. Going from a starter on team with average starters and replacing him with a reliever from the team with the worst relievers would get about a half run per 9 innings (on average), or about 0.05 runs per inning. The average pitches per inning (in 2011) was 16.3, so to eat an entire inning of the pitch count would take at least 16 pitches (that 16.3 is a bit rough since it includes starters and relievers; if there's a better number, feel free to post it). The average pitches per AB was 3.8. So in the favorable case, where you're moving from an average starter to a bad reliever, forcing about 20 pitches in one AB (16.3 more than average) would earn about 0.05 runs.
Getting 0.05 runs per AB would net 30 runs over a season of 600 PAs (for a WAR of about -8 for this hypothetical SS, best I can tell). So eating an entire inning of pitches every AB, even if you were bringing in a bad reliever for an average starter every time, isn't enough.
In essentially a best-case scenario, taking the average starter from the best team and replacing him with the average reliever from the worst team is worth just over 1 run per 9 innings, so about 0.1 runs per inning. That still only produces around 60 runs for a full season (around -5 WAR).
To produce all 93 runs for this SS to get to replacement, in 600 fouling-off PAs, the non-hitter would need 0.155 runs per PA, which would, best-case, be about 1.5 innings worth of pitches in addition to the 3.8 of the average hitter -- about 29 pitches per PA. In the more realistic (?) case of great pitcher to bad reliever, it would take 50+ pitches per AB to reach replacement level.
After all this, what I really wanted to know was how much taking a bunch of pitches was worth. I would say that for every 1 point differential in FIP/ERA between starter and reliever, each pitch per AB more than 3.8 is worth approx 0.0068 (1 ERA point / 9 innings / 16.3 pitches per inning) runs. So even a 14-pitch AB isn't worth as much as a tenth of a run.