The major difficulty for the cult of saints as it pertained to both Jesus and the Virgin Mary is that you can't claim any physical relics existed, as both did not leave their bodies on earth, but ascended into heaven. Thus, the two figures at the top of the community of saints couldn't be incorporated into common praxis. But people compensated for this by claiming relics that either touched Christ, such as the Cross or the Shroud of Turin, or were body parts that would have been left behind. Christ's baby teeth, blood, and foreskin were claimed as relics at various times, as were vials of Mary's tears or breast milk. The latter almost certainly contributed to the profusion of lactation miracles that we see in later medieval sanctity.

I don't know of any references to Christ's urine or feces as relics. My knowledge is not, of course, comprehensive, but as the negative associations of human effluvia are at least as old as the written record and there were so many other more valid and holy-seeming remains that could be used, I would be fairly confident in arguing from this absence. But there are scattered references to holy turds, as it were, such as this one from the Irish annals:

Once, when Áed, not yet king, came through Othan Muru, he washed his hands in the river that goes through the middle of the town. (Othna is the name of the river, and from it the town—i.e., Othna—is named.) He took a handful of water to put on his face. One of his men stopped him: ‘O king,’ he said, ‘do not put that water on your face.’ ‘Why?’ asked the king. ‘I am ashamed to say,’ said he. ‘What shame do you have at telling the truth?’ asked the king. ‘This is it,’ he replied; ‘the clergy's privy is over that water.’ ‘Is it there,’ asked the king, ‘that the cleric himself goes to defecate?’ ‘It is indeed,’ said the youth. ‘Not only,’ said the king, ‘shall I put it upon my face, but I shall also put it in my mouth, and I shall drink it (drinking three mouthfuls of it), for the water into which his faeces go is a sacrament to me.’

That was told to Muru, and he thanked God that Áed Alláin had such faith. Then he summoned Áed Alláin to him (Áed Uaridnach was another of his names), and Muru said to him: ‘Dear son,’ he said, ‘as reward for that reverence you have given the church, I promise, in God's witness, that you will take the kingship of Ireland shortly...

But note that even here, it is not the feces, but rather Áed's humility and devotion, that are being singled out for mention.

The book to read on Christ's blood and other similar relics is Caroline Walker Bynum's Wonderful Blood: Theologies and Practice in Late Medieval Northern Germany and Beyond (Philadelphia: University of Pennsylvania Press, 2007), or anything else Bynum has ever written, really.