Browns spokesman Zak Gilbert said Weeden showed symptoms postgame, was examined and diagnosed.
Gilbert could not say when Weeden might have been hit in the head, or when the injury happened.
Tanney was on the Dallas Cowboys' practice squad before the Browns signed him this past Tuesday.
Weeden had an up-and-down game against Jacksonville, throwing for 370 yards, three touchdowns, two interceptions and a lost fumble.
The three turnovers all occurred in the final 1:19 of the first half, which gave the Jaguars 13 points.