The Papua New Guinean was replaced during the second half of the victory at Salford Red Devils and left the field clutching at his right shoulder, with Toby King replacing him.
After the game, Burgess said the injury appears to be a torn pectoral muscle which, if confirmed, could keep Tai out for the remainder of the season.
With his contract with the club nearing its end, that would naturally bring fears that Tai has played his final game for the club but for now, Burgess is focused on getting a firm diagnosis.
“It’s a suspected torn pec, so that’s not good news for Rod,” he said.
“He’s been really consistent for us. We’ll scan it and confirm it, but that’s what it looks like.”
Warrington also came into the game without hooker Sam Powell and with Danny Walker still absent, Jordy Crowther and Stefan Ratchford were forced to share dummy-half duties at the Salford Community Stadium.
Burgess confirmed Powell was a late withdrawal during Thursday’s captain’s run but is hopeful he will be available to face Catalans Dragons next weekend.
“We lost Sam yesterday in training – it certainly wasn’t a planned thing,” he said.
“He’s had a bit of a niggle and it didn’t pull up too well yesterday.
“Hopefully he’s good for next week – he’s just had a bit of a grumble and pulled up a bit sore.”