Burns, 25, has appeared in five games with Toronto this season, going hitless in five plate appearances with two stirkeouts. He has spent the majority of the season with Buffalo, hitting .238/.298/.350 with two home runs, 16 RBI, 18 walks and 34 strikeouts in 225 plate appearances.
Girodo, 25, will provide the Blue Jays with a left-handed reliever out of the bullpen as Toronto shifts to NL rules for the next two games against the Philadelphia Phillies.
He has a 4.32 ERA with four strikeouts, two walks and three home runs allowed in 8.1 innings with Toronto this season.
The Blue Jays, who moved into the second wild card position on Tuesday, take on the Phillies Wednesday night at 7:05 p.m. ET.