No. We know that because Harry DID touch Nagini with no stated ill effects on her.
This happened in HP7 ("The Deathly Hallows"), when Harry and Hermione went to Godric's Hollow and Harry went into Bathilda Bagshot's house; Nagini tried to detain him before Voldemort's arrival by physically restraining him.
He could not get enough breath into his lungs to call back: Then a heavy smooth mass
smashed him to the floor and he felt it slide over him, powerful, muscular—
“No!” he gasped, pinned to the floor.
“Yes,” whispered the voice. “Yesss … hold you … hold you …” “Accio … Accio Wand …”
But nothing happened and he needed his hands to try to force the snake from him as it coiled itself around his torso, squeezing the air from him...
The bolded part of the quote shows that he was actually touching the snake with bare hands, in case someone tries to lawyer out of this by saying that it has to be skin to skin contact.