Spokane High School in Mead, Washington,[1] and later attended the Oregon State University where he played college baseball for the Beavers.

[7][8] Rasmussen spent his first professional season in 2019 with the Single–A Wisconsin Timber Rattlers, High–A Carolina Mudcats, and Double–A Biloxi Shuckers.

On May 21, 2021, the Brewers traded Rasmussen and J. P. Feyereisen to the Tampa Bay Rays in exchange for Willy Adames and Trevor Richards.

[12] In 2022, Rasmussen was named American League Pitcher of the Month for August, when he registered a 3–1 record in six starts to go along with a 1.57 ERA and 33 strikeouts against only four walks.

Rasmussen began the 2023 season out of the Rays' rotation, making 8 starts and registering a 4–2 record and 2.62 ERA with 47 strikeouts in 44.2 innings pitched.

[14][15] On July 8, Rays manager Kevin Cash stated that Rasmussen would undergo an internal brace procedure on his elbow and miss the remainder of the season.