Franklin entered the game as a defensive replacement in the third inning but departed after playing two innings two days after being hit in the head by a teammate's bat while on deck.
Franklin was not in the starting lineup against Rangers left-hander Cole Hamels, but he went into the game when center fielder Kevin Kiermaier left after the second inning. The team said Kiermaier was bothered by left hip tightness.
Kiermaier's status will be evaluated Saturday.
Franklin flied out in the third inning.
"I thought everything was fine," Franklin said. "For some reason, right after my at-bat, after I ran, I had cotton mouth. Felt like when I went out to left, I felt like I could just pass out there. It was kind of a really weird feeling."
"He came out of the game feeling symptoms kind of like what he felt the other day," Rays manager Kevin Cash said. "Lightheaded, tired, kind of exhausted feeling."
Kiermaier smacked the back of Franklin's head with a bat when Franklin bent to grab some pine tar in the on-deck circle before leading off the first inning Wednesday against San Diego. Kiermaier was stretching a few feet in front of Franklin, swinging the bat over his head with one arm. Franklin was wearing a helmet.
Franklin drew a walk and scored in the first inning Wednesday before leaving with a head bruise. He passed concussion tests Wednesday and Friday.
Tampa Bay had a day off Thursday.
The Associated Press contributed to this report.