Power derived from souls is required
The Fiendish Codex 2 from 3.5e includes a lot of information regarding promotions, including on page 11:
Second, promoting a devil from one form to another requires a considerable expenditure of the divine magic harvested from the breakdown of corrupt souls.
Further, it is also mentioned on page 12 that this also affects demotions:
Promoting an underling uses up a supply of the divine, soul-derived energy on which Baator runs. Demotions, on the other hand, free up a quantity of that corrupt force.
Page 9 of the same source contains details on the process of extracting the required energy from the souls:
(...) the soul shells undergo an awful program of torture, the gruesome details of which are best left to the imagination. While slowly peeling away every last iota of the petitioner’s individuality, the process releases magical energy, which flows to the local lord as specified in the Pact Primeval. Torture teams composed of chain-slinging kytons and masked pain devils mercilessly terrorize and mutilate the souls of the damned until every scintilla of extractable magic has been wrung from them.
(...) the quivering husk that remains is dropped into a processing crater, such as Avernus’s dreaded Maggot Pit. There it quickly perishes and is reborn as a lemure, the lowliest of the various devil forms.
"Local lord" in this passage refers to the Lords of the Nine Hells which are also mentioned in the 5e Monster Manual.
Some references related to promotion can be found in MToF. The Three Paths to Power1 section in particular includes some information on how the different devils are chosen for promotion, which can be considered a form of constraint or limitation for promotions:
Rank-and-file devils have three ways of ascending through the ranks. The Nine Hells uses a complex system of rules to quantify and recognize a devil’s deeds.
The three paths are Souls (receiving credit for collecting mortal souls), Glory (receiving credit for slain foes) and Treachery (replacing a superior by getting them killed)
The Monster Manual entry on devils, under Infernal Hierarchy1 also refers to a constraint on promoting only certain devils (and not just any lower-ranked one):
No devil can promote or demote another devil that has not sworn fealty to it, preventing rival archdevils from demoting each ther's most powerful servants. Since all devils swear fealty to Asmodeus, he can freely demote any other devil, transforming it into whatever infernal form he desires.
Unfortunately, I was unable to find any information from 5e specifically detailing an actual resource being consumed for promotion, as the references above from 3.5e. I searched the Monster Manual and the Blood War section on MToF.
Note that in the case of demons, the Monster Manual does mention that demon lords expend "considerable magical power" to promote lesser demons but no such specific language is used for devils.
1. As suggested by illustro