Very little, though a few have connections to it.
Of the Great Old One patrons listed in the Player's Handbook and Sword Coast Adventurer's Guide, none have a solid interest in the Blood War.
According to Elder Evils (2007), Zargon was once ruler of the Nine Hells in ancient times, before he was driven out by Asmodeus. However, following his exile to the Material Plane, he has long since given up his plans to return to the Nine Hells, deciding to take the Material Plane instead:
The slime bred new creatures that Zargon sought to corral and use as foot soldiers to reclaim his place in the Nine Hells. But as his power grew, Zargon realized this world could be his. He had no reason to return to his native plane when he could claim this one.
In the Forgotten Realms, Moander, listed as a Great Old One in the Sword Coast Adventurer's Guide, actually resided on the 223rd layer of the Abyss, Offalmound. However, Moander was slain, and according to Player's Guide to Faerûn p.165, this caused his realm to be destroyed or expelled from the planes, meaning that Moander no longer has any direct stake in the Blood War.
Kezef the Chaos Hound, one of the Seven Lost Gods according to the Forgotten Realms Campaign Guide (2008), hunts many deities, but has no particular connection to the Blood War.
Dendar the Night Serpent, also one of the Seven Lost Gods, is said in Powers and Pantheons (1997), p. 188, to be to be an ancient elder evil, synonymous with Nidhogg of Norse myth, the dragon who gnaws on the world tree Yggdrasil. She inhabits the Gray Waste of Hades, which is used as a battlefield of the Blood War, but Dendar usually does not leave her lair, content to feast on the nightmares of mortals, and has no particular interest in the Blood War which rages outside her cave.
Ghaunadaur is enemy of Lolth, whose realm the Demonweb Pits was a layer of the Abyss, but by current Forgotten Realms timeline her realm is separate from the Abyss. He now occupies a separate plane called the Dismal Caverns or Deep Caverns, and has no known connections to the Blood War.
Tyranthraxus has possessed mortal bodies on the Material Plane and desired to conquer Faerûn, but has no known connections to the Blood War. According to Curse of the Azure Bonds, his origin is unknown and "none of the lower planes will claim him".
Tharizdun of the Greyhawk lore is an ancient entity of utter bleakness and annihilation, imprisoned by the gods to prevent him from destroying the entire multiverse. He has no connection to the Blood War. He may have a tangential connection via the Elder Elemental Eye, a proxy identity under which he was revered in the Temple of Elemental Evil, and which brought together the power of evil elemental princes who may themselves have some small stake in the Blood War.
Great Cthulhu was omitted from most D&D books for copyright reasons, so there is unlikely to be lore connecting him to the Blood War. Traditionally, he sleeps in his forgotten city beneath the oceans.
The Player's Handbook ends its list with "other unfathomable beings", which suggests that some other Great Old One patrons might exist who care about the Blood War. However, they are described in the Player's Handbook as having motives "incomprehensible to mortals", and the general trend is that Great Old Ones tend to be bizarre entities even older than the war between demons and devils.