I wasn't able to find much lore regarding effects of holy symbols apart from "allows divine spellcasters to channel deity energy" and "require blessing from associated church to work". The logic seems to be that the symbol acts as a target (i.e. casting "focus") to direct the divine magic being channeled, and must therefore be paired with the deity like a bluetooth device to function.
What would happen if an underground church of an evil god went around carving the god's emblem into the forehead of everyone they could find, blessing it each time?
There is precedent that gods have agency of their own and can perform varying levels of action upon mortals without needing assistance from divine spellcasters, so would this allow the god to channel divine magic into/through these people? Would it simply make them easier to target?
What would be the difference if the symbol had instead been applied to a clumsy peasant tripping forehead-first onto a badly (or well) placed red-hot branding iron for the symbol? In this case what if the inadvertently branded symbol was later secretly blessed by the relevant church?
In the case of a birthmark matching the holy symbol, it was not created specifically to be the symbol but matches coincidentally. Does this differ from an application of the actual symbol, accidental or intentional?
To be very specific, I'm asking about any implications that would differentiate well known holy symbols widely associated with a particular deity from any other symbol (or quasi-religious icon) which could be constructed in an identical manner but lacks any preexisting association with a god. This is not about mundane or social effects like identifying membership in a certain group.
Aside from being used as a spellcasting focus, the 5e sourcebooks indeed describe additional effects of a holy symbol:
It has to be used when activating features like Turn Undead or Turn the Unholy.
There are spells like Spirit Guardians which explicitly uses a holy symbol as a material component.
Channel Divinity: Preserve Life also explicitly requires a holy symbol.
Some monsters like Jiangshi are susceptible to holy symbols and can't touch them. Probably that's why a Monster Hunter's Pack has it (both are from Van Richten’s Guide to Ravenloft).
So yes, an actual symbol of a deity has some kind of innate power, which can manifest in various effects. The exact mechanism is unknown — gods work in mysterious ways.
DM can also freely introduce additional effects, which can occur in your games. Maybe a holy symbol repels lesser fiends, is used for blessing the land, or emits light when being invoked. The latter example actually is from the PHB (emphasis mine):
However, many specific things which a holy symbol can (or can not) do are up to the DM. There is no strict canon for this, so you still should ask your DM. This is also applied to cases like a holy symbol being accidentally branded — some people would see this as a sign, while others would call this a blasphemy.