The former civil servant and academic said he was "very happy" at Channel 4, where he has been since 2009.
He has been among the names mentioned as a possible replacement for Lord Patten, who announced he was standing down "on the advice of my doctors" after having heart surgery.
Lord Burns told reporters: "Being chairman of the BBC Trust I don't think is one of the things that's still on my list of ambitions. That's a rather shrinking list, as you can imagine. I'm very happy doing what I am doing."
Channel 4's annual report, which was published yesterday, shows the broadcaster had an "operating deficit" of £15 million in 2013 but was "positioned to deliver a break-even operating result" this year. Its total audience share, across all its channels, fell from 11.5 per cent to 11 per cent in 2013.