Heimerdinger, Inventor has all [T] abilities of all friendly units, including Renata Glasc, Mastermind, which has the text saying her abilities can only be used at a battlefield. Does Heimerdinger need to be at a battlefield to use Renata Glasc, Mastermind's [T] abilities?

No, Heimerdinger, Inventor does not need to be at a battlefield to use Renata Glasc, Mastermind's exhaust abilities.

According to the RiftJudge FAQ (Ref: 650), while Heimerdinger inherits the activated exhaust ability from Renata Glasc, Mastermind, he does not inherit her passive battlefield restriction. He is free to activate that copied ability from his base location.

Important Note on Exhausting: While Heimerdinger gains access to these abilities, remember that he still functions as a single source for them. When you exhaust Heimerdinger to pay the cost of one of these copied abilities, you choose only one specific ability to activate, not all of them simultaneously (Ref: 8927, 1044).