There is no hard-and-fast rule in situations like these if you want to be successful long-term. Saying you should *always* keep the guy in the minors is no more correct than saying you should *always* promote the guy if it increases your playoff chances by any small degree. You have to look at the circumstances on a case-by-case basis.
Here, you have the question of whether Darwin Martin is a sufficient upgrade over Chad Butler and how likely whatever difference exists is likely to increase your odds of making the playoffs. Weighed against whatever that increased likelihood, you have to consider the future cap implications of having multiple high OVR players become arb-eligible together.
I am going to assume you aren't facing fatigue issues at the ML level, so the difference between Martin and Butler is purely an issue of effectiveness. So below (for formatting purposes) are the two ratings lines -
There's no question that Martin is better, right now, but how much of an improvement over Butler can the OP reasonably expect (and is it worth the $ issues)? Butler has a 1.20 WHIP and 3.74 ERA so far this season, certainly nothing too shabby. Over the remaning 70 games, in the 4-man rotation being run, whoever has the rotation spot will make 17-18 starts. For comparison purposes, have the starter everage between 6 and 7 IP per start. Further assume Martin does well and posts the same ML ERA that he has in his minor league career (a BIG assumption) - he would have an ERA of 3.11.
With those assumptions, how will the two SP compare over the rest of the regular season (projected ~113 IP, reached by 17.5*6.5)?
Butler, at 3.74, would allow ~47 ER. Martin, at 3.11, would allow ~39 ER. An 8-run difference, spread out over 17-18 starts.
OP, your team is underperforming at the moment, with a .537 Expected Winning Percentage and only a .495 Actual Winning Percentage. Your pitching staff, particular your startin pitching has not been a problem. Where others teams seem to be performing better than you is offense and closing out game.
With respect to the latter, your 8-14 record in 1-run games (no doubt hurt by the performance of Louie Olmedo and his 1.53 WHIP, 5.91 ERA, 2W-7L, 14/20 saves record). If I am looking at your budget/payroll correctly, your players current cost $44.1M. Even after transfers into prospect budget, your player budget is $59M. I'd explore taking on the expiring high-priced contract of a veteran closer/setup A or difference-making bat before I promoted Martin. You may be able to pick up most, if not all, of the 8 runs he projects to save without taking on any financial obligations beyond this season and not giving yourself a headache in 3 seasons when you are paying $40M+ out in arb in one year.
Of course, things change if you expect Butler to post a 5.50 ERA or Martin to post a 2.50 ERA, but even then you make the determination on a case-by-case basis - don't get wed to "rules".