Contradiction with Mark 7:15
Exodus 28:34 emphasizes the importance of specific external garments, while Mark 7:15 suggests that external things are not what make a person unclean or defiled.
Mark 7:15: If any man have ears to hear, let him hear.
Contradiction with Romans 14:14
Exodus 28:34 concerns prescribed external holy garments, whereas Romans 14:14 says that nothing in itself is unclean, which could include garments or external objects.
Romans 14:14: I know, and am persuaded by the Lord Jesus, that [there is] nothing unclean of itself: but to him that esteemeth any thing to be unclean, to him [it is] unclean. [unclean: Gr. common]
Contradiction with Colossians 2:16-17
Exodus 28:34 describes specific ceremonial attire, but Colossians 2:16-17 suggests that such ceremonial matters are just a shadow of things to come, indicating they are not fundamentally important.
Colossians 2:16-17: Let no man therefore judge you in meat, or in drink, or in respect of an holyday, or of the new moon, or of the sabbath [days]: [in meat...: or, for eating and drinking] [respect: or, part]
Contradiction with Hebrews 9:10
Exodus 28:34 maintains the importance of ritual, while Hebrews 9:10 implies those rituals and regulations were meant only until the new order, suggesting their temporal nature.
Hebrews 9:10: [Which stood] only in meats and drinks, and divers washings, and carnal ordinances, imposed [on them] until the time of reformation. [ordinances: or, rites, or, ceremonies]