[1][2] He was selected by the Tampa Bay Rays in the 24th round of the 2009 Major League Baseball Draft, but did not sign, opting to enroll at Oklahoma State University, where he played college baseball for the Oklahoma State Cowboys, competing in the Big 12 Conference of the National Collegiate Athletic Association's (NCAA) Division I.

[8] Heaney began the 2014 season with Jacksonville, and was promoted to the New Orleans Zephyrs of the Triple-A Pacific Coast League in May.

In his Triple-A debut, he faced the Oklahoma City Redhawks and went five innings, allowing one run on seven hits while adding seven strikeouts and no walks.

[10] In 7 games (5 starts) for Miami during his rookie campaign, Heaney posted an 0-3 record and 5.83 ERA with 20 strikeouts across 29+1⁄3 innings pitched.

[11] On December 10, 2014, the Marlins traded Heaney to the Los Angeles Dodgers, along with Chris Hatcher, Austin Barnes, and Enrique Hernández, in exchange for Dan Haren, Dee Gordon, and Miguel Rojas.

On April 30, Heaney received a platelet-rich plasma injection in his left elbow, sidelining him for 6 weeks.

[16] On June 28, Heaney underwent evaluations on his elbow and found no improvement, pushing back his return even further.

[15] On June 5, 2018, the day of his 27th birthday, Heaney pitched a one-hit complete game shutout as the Angels won 1–0 over the Kansas City Royals.

[15] Heaney entered 2019 as the projected #1 starter but suffered an elbow inflammation and was shut down after his first spring start.

[23] In the 3–7 no-decision loss to the Oakland Athletics, Heaney pitched 4+2⁄3 innings, striking out 6 batters and allowing 1 run.

[15] On July 30, 2021, Heaney was traded to the New York Yankees in exchange for Janson Junk and Elvis Peguero.

[24] On August 12, Heaney was the starting pitcher in the first Field of Dreams game in Dyersville, Iowa against the Chicago White Sox.

[26] On November 10, 2021, Heaney signed a one-year, $8.5 million contract with the Los Angeles Dodgers.

In Game 4 of the 2023 American League Championship Series, Heaney lasted only 2⁄3 of an inning, surrendering four hits, three earned runs, and one walk.

Heaney made 32 appearances (31 starts) for Texas during the 2024 campaign, compiling a 5-14 record and 4.28 ERA with 159 strikeouts across 160 innings pitched.