Vice-captain Burrell has signed a new long-term contract, while winger Diamond's loan has been extended until the end of the season.
Sheffield-born Burrell is in his second spell at the club, having rejoined in 2016. He has made over 180 appearances for Town.
The defender scored a hat-trick against Aldershot last season.
Diamond has made 12 appearances so far this season, creating three goals, and manager Simon Weaver has described the 19-year-old as “a bit of a throwback winger.
“He wants to get at the full-backs and he’s exciting to watch. Plus, he’s got that good raw edge about him and determination to win.”
It is Burrell's determination his manager also pointed to.
“He’s an incredible athlete and he wears his heart on his sleeve and plays for the badge game in, game out,” said Weaver, who has been named as the Conference's manager of the month for September. “He plays with knocks, plays in different positions and I can’t think more highly of him.”