Deducing why Duryodhana had desired Shalya in the first place, Yudhishthira demanded from Shalya that he would demoralize Karna and dampen his spirits when he eventually became Karna's charioteer. Shalya promised to do so
Yes, he did. Here is what this chapter of the Udyoga Parva of the Mahabharata says:
And Yudhishthira said, ‘O valiant king, it has been well-done by thee that being pleased at heart thou hast plighted thy truth to Duryodhana. But good betide thee, O ruler of the earth, I ask thee to do one thing only. O king, O best of men, thou wilt have to do it solely for my sake, though it may not be proper to be done. O valiant one, hear what I submit to thee. O great king, thou art equal to Krishna on the field of battle. When, O best of kings, the single combat between Karna and Arjuna will take place, I have no doubt thou wilt have to drive Karna's car. On that occasion, if thou art inclined to do good to me, thou must protect Arjuna. O king, thou must likewise so act that the Suta's son Karna may be dispirited and the victory may be ours. Improper it no doubt is; but, O my uncle, for all that thou must do it.’
Salya said, 'Good betide thee. Listen, O son of Panda. Thou tellest me to so act that the vile son of the Suta may be dispirited in fight. To be sure, I shall be his charioteer on the field, for he always considers me equal to Krishna. O tiger like descendant of Kuru, I shall certainly speak to him, when desirous of fighting on the field of battle, words contradictory and fraught with harm to him, so that bereft of pride and valour, he may be easily slain by his antagonist. This I tell thee truly. Asked by thee to do it, this I am determined to do, O my son. Whatever else I may be able to bring about, I shall do for thy good. Whatever troubles were suffered by thee together with Draupadi on the occasion of the game at dice, the rude inhuman words uttered by the Suta's son, the misery inflicted by the Asura Jata and by Kichaka, O illustrious one, all the miseries experienced by Draupadi, like those formerly experienced by Damayanti,--will all, O hero, end in joy. Thou shouldst not be aggrieved at this; for Destiny is all powerful in this world; and, O Yudhishthira, high-minded persons have to endure miseries of various kinds, nay, even the gods themselves, O king, have suffered misfortunes. O king, O descendant of Bharata, it is narrated that the high-minded Indra, the chief of the celestials, had to endure together with his wife very great misery, indeed.'
And Shalya followed through on this promise. Here is an example of his demoralizing words to Karna from the Karna Parva of the Mahabharata:
When keen-pointed shafts winged with Kanka feathers, shot by Phalguna of mighty arms and impelled from his bow-string and sped with all his energy will seek thee then wilt thou lament thy encounter with that hero. When Partha, called also Savyasaci, taking up his celestial bow, will scorch the (Kuru) army and afflict thee exceedingly with keen shafts, then, O Suta's son, wilt thou repent (of thy folly). As a child lying on the lap of its mother seeks to seize the Moon, even so dost thou from folly seek to vanquish the resplendent Arjuna stationed on his car. In desiring, O Karna, to fight today with Arjuna of keen-edged feats, thou art for rubbing all thy limbs against the keen edges of a trident. This thy challenge of Arjuna, O Suta's son, is like that of a foolish young little deer of activity challenging a huge lion excited with wrath. Do not, O Suta's son, challenge that prince of mighty energy like a fox gratified with meat in the forest challenging the maned monarch of the forest. Do not be destroyed, encountering Arjuna. Thou, O Karna, challengest Dhananjaya, the son of Pritha, even like a hare challenging a mighty elephant with tusks large as plough-shafts, and with the juice issuing out of its mouth and rent cheeks. From folly thou art piercing, with a piece of wood, the black cobra of virulent poison excited to fury within its hole, in desiring to fight with Partha. Endued with little understanding, thou, O Karna, disregarding that lion among men, viz., the son of Pandu, yellest at him, like a jackal that, disregarding a maned lion excited with wrath, yells at him. As a snake, for its own destruction, challenges that foremost of birds, viz., Vinata's son, possessed of beautiful plumage and great activity, even so dost thou, O Karna, challenge Dhananjaya the son of Pandu. Thou desirest to cross without a raft the terrible ocean, the receptacle of all the waters, with its mountain waves and teeming with aquatic animals, when at its height at the rise of the Moon. O Karna, thou challengest Dhananjaya, the son of Pritha, to battle even like a calf challenging a smiting bull of keen horns and neck thick as a drum. Like a frog croaking at a terrible and mighty cloud yielding copious showers of rain, thou croakest at Arjuna who is even like Parjanya among men. As a dog from within the precincts of the house of his master barks at a forest-roaming tiger, even so, O Karna, thou barkest at Dhananjaya, that tiger among men. A jackal, O Karna, residing in the forest in the midst of hares regardeth himself a lion till he actually sees a lion. Even so, O son of Radha, thou regardest thyself a lion, for thou dost not behold that repressor of foes, that tiger among men, viz., Dhananjaya. Thou regardest thyself a lion till thou beholdest the two Krishnas stationed on the same car like Surya and Candramas. As long as thou dost not hear the twang of Gandiva in great battle, so long art thou able to do what thou pleasest. Beholding Partha, causing the ten points of the compass to resound with the roar of his car and the twang of his bow, and beholding him roaring like a tiger, thou wilt become a jackal. Thou art always a jackal, and Dhananjaya always a lion. O fool, in consequence of thy envy and hatred for heroes, thou always, seemest to be like a jackal. As a mouse and a car are to each other in strength, or a dog and a tiger, a fox and a lion, or a hare and an elephant, as falsehood and truth, as poison and nectar, even so art thou and Partha known to all by your respective deeds.
Before the start of the battle, Yudhishthira asked Shalya to bless him with victory in the war.
-Hearing, O king, these words of Gautama, and paying him due honours, the king proceeded thither where the ruler of the Madra was. Saluting Salya and walking round him the king said unto that invincible warrior those words that were for his own benefit.
'Yudhishthira said,--'Obtaining thy permission, O invincible one, I will fight without incurring sin, and permitted by thee, O king, I will vanquish (my) valourous foes.'
Shalya asked how he can help in circumstances when he is against him in the war and here, Yudhishthira told that he should weaken Karna.
"Salya said, 'Under these circumstances, say, O best of kings what aid shall I render thee? I shall, of course, fight for the sake of (thy) enemy, for I have been made one of their party by the Kauravas with their wealth.
"Yudhishthira said, 'Even that is my boon, O Salya, which was solicited by me during the preparations (for the fight). The energy of the Suta's son (Karna) should be weakened by thee in battle.'
Shalya then promised him that he would accomplish his wish.
"Salya said, 'This thy wish, O Yudhishthira, shall be accomplished, O son of Kunti. Go, fight according to thy pleasure. I shall look after thy victory."
In Karna Parva, Shalya did demoralize by listing the feats of Arjuna.
Why didst thou not slay Partha at Virata's city when thou hadst the advantage of being protected by Drona and Drona's son and Kripa and Bhishma and the other Kauravas? There where, like a pack of jackals defeated by a lion, ye all were defeated with great slaughter by the diadem-decked Arjuna, what became of your prowess? Beholding also thy brother slain by Savyasaci, in the very sight of the Kuru heroes, it was thou that didst fly away first. By the skirts also of the dvaitya lake, O Karna, when thou wert assailed by the Gandharvas, it was thou that, deserting all the Kurus, didst first run away. Having vanquished in battle the Gandharvas headed by Citrasena, with great slaughter, it was Partha, O Karna, that liberated Duryodhana with his wife. Rama himself, O Karna, before the kings in the (Kuru) assembly spake of the great prowess of both Partha and Keshava. Thou didst frequently hear the words of Drona and Bhishma, speaking in the presence of all the kings, that the two Krishnas are unslayable. I have told thee a little only regarding those matters in which Dhananjaya is superior to thee like the brahmana who is superior to all created beings. Soon wilt thou see, stationed on that foremost of cars, the son of Vasudeva and the son of Kunti and Pandu.