This may be specifically an American slang term, but yes, we commonly say "don't sweat it." It can mean either "you're welcome" or "don't worry about it." Please note that this is VERY informal.
EXAMPLE:
"Hey, thanks for the ride home."
"Don't sweat it."
EXAMPLE:
"I'm worried about our math test next week."
"Seriously? Don't sweat it. You're great at math."
A similar saying, also VERY informal, is saying "no sweat," which means "easily" or "that was easy." It can also sometimes be used in reponse to "thank you," but it doesn't mean "you're welcome." It's kind of how sometimes you might say, "There's no need to thank me. It was easy / It was no big deal." You can also say "no sweat" to gloat after winning a game of any sort, even if it's not a physical sport in which sweating is common.
EXAMPLE:
"Thanks for forwarding that email to me."
"No sweat."
EXAMPLE:
That guy is a chess wizard. I watched him beat 10 people in a row, no sweat.
EXAMPLE:
"Dang, you beat me again..."
"No sweat!"