Dempsey began karting at age eight, following in the footsteps of his father, Cliff Dempsey, a longtime junior formula team owner whose Cliff Dempsey Racing has won multiple Formula Ford Festival and Walter Hayes Trophy events. Peter made his competitive debut in Formula BMW UK in 2004, finishing 22nd. In 2005 he entered a full Formula Ford season, winning 33 of the 43 races he contested and earning the Motorsport Ireland Young Racing Driver of the Year award. He took third overall in the Formula Ford UK championship in 2006 and won the prestigious Walter Hayes Trophy that year, then repeated the win in 2007 alongside another BRDC Golden Helmet. Winning the Walter Hayes Trophy three times placed him alongside only a handful of drivers in the event's history.
In 2008 Dempsey moved to the United States to compete in the Star Mazda Championship with Andersen Racing, winning four races and finishing third in the standings. He also made two starts in the F2000 Championship Series that year. In 2009 he split the season between AIM Autosport and Juncos Racing, taking five wins and finishing second in the championship after a DNF in the final round at Laguna Seca.
Dempsey moved up to the Firestone Indy Lights series in 2011 with O2 Racing Technology. That team withdrew mid-season following controversy at the Milwaukee Mile, and Dempsey signed with Andretti Autosport to complete the year, finishing tenth in the championship. In 2012 he returned with Younessi Racing for the Freedom 100 and the remainder of the season, but that team also departed after Detroit. Belardi Auto Racing signed him for the final five rounds, producing three top-five finishes and an 11th-place championship result across seven starts.
Dempsey stayed with Belardi for a full 2013 campaign. On 24 May 2013 he won the Freedom 100 at Indianapolis Motor Speedway in a four-wide finish decided by 0.0026 seconds — at the time the closest finish in Indianapolis Motor Speedway history. He described holding back in the draft and only committing to the outside run when the three leaders went three-wide and lost their slipstream. He finished fifth in the 2013 Indy Lights championship with 360 points.
Following his driving career, Dempsey served as a driver instructor and assistant engineer at Juncos Racing from 2010 to 2017, ultimately becoming lead engineer and working alongside Ernesto Gonella when the team won the Indy Lights title in 2017 with Kyle Kaiser. He also spent a year as engineer at DEForce Racing in 2017–2018.
Dempsey founded Turn 3 Motorsport in 2016 with his wife Mandy Dempsey, initially as a driver coaching business based in Mundelein, Illinois. The team competed in the Radical Cup North America starting in 2019, winning both driver and team championships across multiple classes that year and overall. Turn 3 also won the NASA 25 Hours of Thunderhill in 2019, leading 638 of 672 laps in only the team's second attempt. From 2020 the team focused on the USF Pro Championships, winning multiple poles and races. In 2024 Turn 3 captured the USF Pro 2000 Championship with driver Lochie Hughes, totalling five wins, five poles, and eight podiums. Dempsey himself won Engineer of the Year at the 2024 Road to Indy awards.
In 2026 Dempsey returned to racing as a bronze-rated FIA driver, joining Racing Team Turkey by TF in the FIA World Endurance Championship, sharing a Chevrolet Corvette Z06 GT3.R with Charlie Eastwood and Salih Yoluc. The entry finished sixth in class at the 2026 24 Hours of Le Mans.