Tessier played major junior in the Ontario Hockey League (OHL) where he served as a team captain for the Ottawa 67's. Establishing his success with faceoffs as a top aspect of his game, he was awarded the OHL Top Faceoff Award in the trophy's inaugural three years through 1998 to 2000, as well as the CHL Top Faceoff Award in 1999. After an 84-point season in 1998–99, Tessier led the 67's to a Memorial Cup championship as the tournament hosts. In his fourth and final year with the 67's, he was named to the OHL First All-Star Team, earned the Dan Snyder Memorial Trophy for his humanitarian efforts and won Leo Lalonde Memorial Trophy as the OHL's best overage player with a second consecutive 84-point season.
Undrafted by a National Hockey League (NHL) club, Tessier earned an invite to the Montreal Canadiens training camp in 2000. He spent his professional rookie season in the East Coast Hockey League (ECHL) with the Louisiana IceGators, recording 35 points in 60 games. The following season, he earned another NHL training camp invite with the Ottawa Senators. Tessier spent the 2001–02 season, however, overseas in the Holland Super League with the Amstel Tijgers where he finished second in league scoring with 43 points in 33 games, as well as first in playoff scoring. Tessier subsequently bounced between the North American minor professional leagues in the American Hockey League (AHL) and ECHL as well as the European leagues, such as the German 2nd Bundesliga (DEL2), and Bundesliga (DEL), the Swiss Nationalliga B (SUI-B), and the British Elite Ice Hockey League (EIHL). In 2004, he competed in the CBC's inaugural season of Making the Cut
, a hockey-based reality show in which contestants compete for a spot on one of the six Canadian NHL teams.