Since joining Persepolis in 2014, Taremi has become the top scorer of the Persian Gulf Pro League on two occasions (2015–16, 2016–17). Taremi has also been a member of the Iran national football team since 2015.
He is the third and the youngest child of his family. His older brother Mohammad Taremi is also a football player. Mehdi has good professional and personal relationships with all national football players but is on especially good terms with Sardar Azmoun, Morteza Pouraliganji and Mohsen Mosalman, the latter being his closest friend and roommate in Persepolis.
Taremi started his career with Bargh Bushehr's Academey before moving on to Iranjavan's Youth Academy. In addition, his physiotherapist is Saeed Shaykhy, who is a world renown physiotherapist in The Netherlands.
He joined Shahin Bushehr in the summer of 2010. He played 7 games and scored one goal in all competitions. In the winter of 2012, he was released by Shahin Bushehr to spend his conscription period in a military club, but he failed to join a club and was forced to spend his conscription period in a usual garrison.
He joined Iranjavan in the summer of 2013 and signed a two-year contract until the end of the 2014–15 season and was given the number 9. Taremi scored 12 times in 22 matches in 2013–14, and became the leagues second top scorer after Mokhtar Jomehzadeh.
After becoming the Azadegan League's top scorer, Taremi had offers from many teams, He joined Persepolis F.C. in the summer of 2014 and signed a two-year contract through June 2016.
He made his debut in 1–1 draw with Naft Tehran, coming as a substitute for Reza Norouzi in 90th minute. He scored his first goal for the club on 15 August 2014 in a 1–0 victory over Zob Ahan. On 8 April 2015 Taremi scored a penalty in Persepolis 1–0 win in the Asian Champions League over Saudi Arabian club Al Nassr. He scored many goals for the club, and provided quality assists to win the Best Striker Award of the 14th Persian Gulf Pro League at the end of the 2014-15 season.
Taremi missed the first week of the new season due to suspension, but in his first game back on 6 August 2015, he scored in a 2–1 loss to Esteghlal Khuzestan. Taremi scored twice in a 2–0 week five win against Foolad. This was Persepolis' first win of the season and lifted them out of the relegation spot. Taremi's good performances in the month of August earned him the Navad player of the month as voted by fans. On 18 December 2015 Taremi scored a brace against Rah Ahan in Persepolis' 2–0 victory. He ended the first half of the league as top scorer with 9 goals in 12 games.
On 4 April 2016 Taremi scored a brace in a 3–2 victory against Siah Jamegan to keep Persepolis in third place. On 12 April 2016 Taremi was in talks to extend his contract to the end of the 2017–18 season. But, eventually he decided not to renew his contract. On 15 April 2016 Taremi scored a brace in Persepolis' historic 4–2 victory over Esteghlal in the Tehran derby. He became league's top scorer with 16 goals at the end of the season, unless his team lost title at the last week.
Before the start of the new season, Taremi announced that he will not extend his contract, in willing to join to a European team. In July 2016, it was reported that Taremi is joined to Süper Lig team Çaykur Rizespor. The move was cancelled after Taremi returned to Iran and signed a two-year contract with Persepolis. He played his first match for Persepolis in 1–0 win over Saipa, replaced Omid Alishah in 77th minute. He also scored his first goal of the season in week 4 against Saba Qom. He scored a brace in 3–1 away win over Sepahan, in match knowing as Iran's El Clasico. At the end of the season, the club celebrated 10th Iran Pro League title and Taremi became top scorer for second consecutive season with 18 goals.
On 8 May 2017, Taremi scored a decisive hat trick against Emirati side Al Wahda in a 4–2 Long win, to help his team qualify for the Round of 16 of the 2017 AFC Champions League. After his good performances, Taremi was linked with Dinamo Zagreb.
He played for Iran students national football team in Turkey tournament and scored 9 goals for the team.
He was called up by Ali Reza Mansourian to participate in the team's training camp in Kish Island in 2013.
He made his debut against Uzbekistan in a friendly match on 11 June 2015. He also came on as a second-half substitute in a World Cup Qualifying match against Turkmenistan on 16 June 2015.
On 3 September 2015, Taremi scored his first senior goals, a brace, in a 6–0 defeat of Guam in a 2018 FIFA World Cup qualifier. On 23 March 2017, Taremi scored the winning and only goal in Iran's victory against Qatar in the final round of World Cup qualifying. He followed this up with another goal on 28 March 2017, in a decisive 1–0 victory against China. On 12 June 2017, Taremi scored the second goal for Iran in a 2-0 win over Uzbekistan to seal the country's first consecutive qualification to two World Cups.As of 12 September 2017.
As of 12 June 2017. Iran score listed first, score column indicates score after each Taremi goal.
Persian Gulf Pro League (1) : 2016–17
Iranian Super Cup (1) : 2017
Iranian Footballer of the Year (2) : 2016, 2017
Persian Gulf Pro League Player of the Season (2) : 2015–16, 2016–17
Persian Gulf Pro League Top Goalscorer (2) : 2015–16 (16), 2016–17 (18)
Persian Gulf Pro League Best Striker of the Season (3) : 2014–15, 2015–16, 2016–17
Persian Gulf Pro League Team of the Season (3) : 2014–15, 2015–16, 2016–17