Dobbs entered the 2015 season as Tennessee’s starting quarterback. He started and appeared in all 12 regular season games and the bowl game. To open Tennessee’s season on September 5, Dobbs recorded 205 passing yards, two passing touchdowns, 89 rushing yards, and one rushing touchdown against the Bowling Green Falcons in a 59–30 win at Nissan Stadium in Nashville, Tennessee. In a 2OT 31–24 loss to the #19 Oklahoma Sooners in the Tennessee 2015 home opener, Dobbs had 125 passing yards, one passing touchdown, 12 rushing yards, and one rushing touchdown. In a 28–27 loss to SEC East rival Florida at Ben Hill Griffin Stadium, Dobbs had a season-high 136 rushing yards and had a 58-yard receiving touchdown thrown by teammate wide receiver Jauan Jennings on a trick play. Dobbs’s touchdown reception against Florida was the first reception by a Tennessee quarterback since Peyton Manning caught a 10-yard pass from running back Jamal Lewis in 1997 against Arkansas. Against the rival #19 Georgia Bulldogs, Dobbs had a season-high 312 yards passing and three touchdowns to go along with 118 rushing yards and two touchdowns. His efforts in the game lead Tennessee to their first win over the Bulldogs since 2009. Against the # 8 Alabama Crimson Tide in their annual rivalry game, Dobbs had 171 yards passing and one passing touchdown in the narrow 19–14 loss at Bryant–Denny Stadium. Against rival South Carolina, Dobbs passed for 255 yards and two touchdowns in the 27–24 home victory. Dobbs lead Tennessee to a 9–4 record, which was the most for the Tennessee program since 2007. The 2015 season was culminated with a 45–6 victory over the #12 Northwestern Wildcats in the 2016 Outback Bowl. In the bowl game, Dobbs had two rushing touchdowns.

Dobbs started playing competitive football when he was five years old. He attended Wesleyan School and then Alpharetta High School. As a senior with the Alpharetta Raiders football team, he threw for 3,625 yards with 29 touchdowns. Dobbs was a three-star recruit by Rivals.com and a four-star by Scout.com. He originally committed to Arizona State University to play college football, but in February 2013, he changed his commitment to the University of Tennessee.

As a true freshman at the University of Tennessee in 2013, Dobbs played in five games with four starts after starter Justin Worley was injured in a 45–10 loss against the #1 Alabama Crimson Tide at Bryant–Denny Stadium. Dobbs came into the game on the road at No. 1 Alabama and completed 5-of-12 passes for 75 yards. He started his first career game against the #10 Missouri Tigers at Faurot Field. He completed 26-of-42 passes for 240 yards in the 31–3 loss, which was the most passing yards in a freshman debut since 2004; Erik Ainge (118) and Brent Schaefer (123) against the UNLV Rebels. After a 55–23 loss to the #7 Auburn Tigers and a 14–10 loss to the Vanderbilt Commodores, Dobbs put together a solid performance against the Kentucky Wildcats at Commonwealth Stadium. In the 27–14 victory, Dobbs threw his first two career touchdown passes and had a 40-yard rushing touchdown. Overall, he completed 72-of-121 passes for 695 yards with two touchdowns and six interceptions and also rushed for 189 yards and a touchdown in his true freshman season.

Dobbs entered the 2016 season as Tennessee’s starting quarterback in his final season of collegiate eligibility. He started and appeared in all 12 regular season games and the bowl game. Dobbs started the season with a solid performance in a home game against Appalachian State. In the 20–13 overtime win, Dobbs had 192 yards passing but fumbled on the goal line; the ball was recovered by teammate and running back Jalen Hurd to give Tennessee the go-ahead score. In the 2016 Pilot Flying J Battle at Bristol, Dobbs three passing touchdowns to go along with two rushing touchdowns. In a 38–28 comeback victory over the #19 Florida Gators, Dobbs had 319 yards passing, four passing touchdowns, 80 rushing yards, and a rushing touchdown to lead the Volunteers to their first win over the Gators since 2004. Against #25 Georgia, Dobbs had 230 yards passing, three passing touchdowns, and a rushing touchdown to win 34–31. Dobbs’s last touchdown was a Hail Mary throw to wide receiver Jauan Jennings as time expired. The winning play is referenced by many as the “Dobbs-Nail Boot”. With the victory, Tennessee was 5–0 with Dobbs as quarterback and ranked as high as top 10 in some polls. In a 2OT 45–38 loss to the #8 Texas A&M Aggies at Kyle Field, Dobbs had a season-high 398 passing yards and one passing touchdown. In addition, he caught a receiving touchdown from Jauan Jennings, his second career receiving touchdown. Dobbs continued solid performances over the rest of the season: he had five touchdowns, 223 passing yards and 190 rushing yards in a 63–37 win over Missouri and 340 passing yards in a 45–34 loss against Vanderbilt at Vanderbilt Stadium. Despite his play, Tennessee faded from their 5–0 start to finish 8–4.

In the final game of his Tennessee career, Dobbs led the Volunteers past the #24 Nebraska Cornhuskers by a score of 38–28 in the 2016 Music City Bowl at Nissan Stadium in Nashville. He had 291 passing yards, one passing touchdown, 11 rushes for 118 yards, and three rushing touchdowns. Dobbs was named the MVP of the game.

On May 22, 2017, the Pittsburgh Steelers signed Dobbs to a four-year, $2.95 million contract with a signing bonus of $554,295.

Dobbs made his NFL regular season debut on October 7, 2018 in a 41–17 Steelers win against the Atlanta Falcons, as on the final play of the game, he kneeled down for a loss of 3 yards.

On November 4, 2018, in a 23–16 Steelers Week 9 victory against the Baltimore Ravens, Dobbs completed a 22-yard pass to JuJu Smith-Schuster, after stepping in for Ben Roethlisberger, who got injured on the previous play. In Week 14, against the Oakland Raiders, Dobbs once again had to step in for Roethlisberger, who had suffered a rib injury. He finished 4-of-9 for 24 yards and one interception in the 24–21 loss. Overall, in the 2018 season, he appeared in five games and went 6-of-12 for 43 yards and one interception.

On September 9, 2019, Dobbs was traded to the Jacksonville Jaguars for a fifth-round pick in the 2020 NFL Draft. Dobbs was traded after Mason Rudolph won the backup job and Jaguars’ quarterback Nick Foles sustained a broken clavicle during the season opener and was subsequently placed on injured-reserve.

On September 5, 2020, Dobbs was waived by the Jaguars.

On September 6, 2020, Dobbs was claimed off of waivers by the Pittsburgh Steelers, his former team.