Do we move the chains after an objectionable conduct penalty?

Not if the foul comes after the Referee has whistled time in. If an objectionable conduct penalty is called after the Referee has whistled time in for any down, the penalty is L10 from the point of last scrimmage with the down repeated, and the distance chains will not be moved (unless a first down is gained from a penalty against Team B).

The Case Book shows the other side: Team A has 2nd down and goal on the Team B 4-yard line, B1 goes offside and the play gains 1 yard, and A2 is then called for objectionable conduct after the ball is dead. The offside penalty moves the ball to the Team B 2-yard line and 1st down. The objectionable conduct penalty is then applied, giving Team A 1st down and 10 from the Team B 12-yard line. The distance chains are moved as long as the objectionable conduct occurred before the Referee has whistled time in for the next play.

Source: Canadian Amateur Rule Book for Tackle Football 2026-2027, Rule 7, Section 4, Article 1 and Case Book play 4-0-4.