It is up to the DM
The rules do not provide specific guidance, so in the end the DM will need to decide.
But the PHB says PCs make death saving throws
The PHB says:
If damage reduces you to 0 hit points and fails to kill you, you fall unconscious.
And then it says:
Whenever you start your turn with 0 hit points, you must make a special saving throw, called a death saving throw, to determine whether you creep closer to death or hang onto life.
But, when the PHB says "you", it means a PC. So, if a shield guardian were a PC, it would make death saving throws. But a shield guardian isn't a PC. Or at least, since nowhere in the rules it says to treat shield guardians as PCs, then a shield guardian isn't a PC unless the DM decides it's a PC, and in that case you're pretty firmly in houserule territory, so the DM decides.
What is a shield guardian? Is it a monster? An NPC?
Is a shield guardian a monster? Well, if you encounter a hostile one, sure. And it definitely has a stat block. Or maybe it's an NPC.
So, do monsters and NPCs get death saving throws?
The PHB goes on to say:
Most DMs have a monster die the instant it drops to 0 hit points, rather than having it fall unconscious and make death saving throws.
although it also adds that there are exceptions, and that:
Mighty villains and special nonplayer characters are common exceptions; the DM might have them fall unconscious and follow the same rules as player characters.
So, if you choose to look at a shield guardian is a monster or an NPC, then it is explicitly up to the DM.
What about Warforged?
After all, they have a creature type of construct, just like a shield guardian. True, but really the deciding factor as to whether a warforged gets a death saving throw is whether its a PC. PCs pretty obviously get death saving throws, but for NPC or monster warforged, it's up to the DM, just like for other NPC or monsters of playable races.
What about guidance in other 5e source materials?
Shield guardians are mentioned numerous times in the 5e materials (Curse of Strahd, Icewind Dale: Rime of the Frostmaiden, Princes of the Apocalypse, Out of the Abyss, Tome of Annihilation, Waterdeep: Dungeon of the Mad Mage), but nowhere does it describe what happens when a shield guardian hits 0.
Okay. What about things that are sort of like shield guardians, like familiars, sidekicks, steel defenders, homunculi, golems?
There are many creatures that bear some (perhaps slight) resemblance in form and/or function to a shield guardian.
When a familiar drops to 0 hit points, "it disappears, leaving behind no physical form." (PHB/Basic Rules)
When a sidekick drops to 0, it "makes death saving throws, just like a player character". (Tasha's Cauldron of Everything)
The rules make no mention one way or the other about death saving throws for the artificer's Steel Defender (Tasha's Cauldron of Everything), the homunculus (Xanathar's Guide to Everything), or the many, many kinds of golems (Basic Rules, multiple other sources).
The DM has to decide
In the end, the DM will have to decide. Either way has consequences. Allowing death saving throws adds some logistic burden, not allowing them might mean that the guardian can't really participate in high-level combat, since once it's below 1/4 to 1/2 hit points, and hitting 0 will render useless, then, the prudent thing to do might be to withdraw it from combat.
And then, of course, what to do with a dead shield guardian? But then, that's another question.