Weems attended McMain High School, but was forced to move before his junior year due to Hurricane Katrina. He transferred to Taft High School, where he was named All-American.
He accepted a scholarship from the University of Oregon. As a junior, he started 7 games. He became a regular starter at left tackle as a senior and was part of an offense that ranked third in the nation in scoring, sixth in total offense and fifth in rushing.
On April 29, 2012, he signed with the Minnesota Vikings as an undrafted free agent. He was waived on July 24.
On July 27, 2012, he signed with the New England Patriots as a free agent. He was released on September 1.
On September 3, 2012, he was signed by the Indianapolis Colts to join their practice squad. He was cut on September 18.
On October 2, 2012, he signed with the Denver Broncos to join their practice squad.
On December 5, 2012, he was signed by the Dallas Cowboys from the Denver Broncos' practice squad and was declared inactive in the last 4 games. The next year, he was declared inactive for all of the regular season games. In 2014, he missed most of the preseason with a shoulder injury and was placed on the injured reserve list on September 17.
In 2015, the team lost Jermey Parnell in free agency and Weems entered training camp as the favorite to earn the swing tackle job. After he struggled in preseason, he was released on September 15, to make room for offensive tackle Charles Brown. On September 17, he was brought back and signed to the practice squad.
On January 13, 2016, he was signed by the Denver Broncos to a future contract. He was released on October 17, 2016.