Burnham announced the decision to approve the drilling of the Jackdaw gas field earlier this week.
Jackdaw is situated 150 miles (241km) east of Aberdeen and is operated by Adura, a joint venture between Shell and Norwegian state energy firm Equinor.
Ministers said the first gas could be produced by the end of this month, helping heat millions of homes, and also announced the creation of a new energy wealth fund.
Environmental campaigners say the decision would hamper efforts to tackle climate change and make "no meaningful difference" to UK energy security.
Based on official forecasts for the amount of gas the UK will need over the next decade, that suggests Jackdaw has the potential to meet around 1.6% of total projected national gas demand between 2026 and 2036.
Operators say it could be connected to the Great British gas network within weeks and could produce gas for the next 11 years.
Trump responded to the decision on Friday, but appeared to confuse oil and gas.
He said: "They're opening up the North Sea oil. I've been after them to do that for two years."
Trump added: "Andy is getting it done, and I guess it's against the liberal principle, but he knows what's right. It's going to make UK a very wealthy country."
Earlier this week, on Wednesday, Trump said the decision was "great news for the wealth and prosperity" of the UK.
Writing on his Truth Social platform, Trump urged the UK to go "full blast" and called the North Sea "one of the greatest and most prosperous areas of oil in the world".
A decision is yet to be taken on whether production can begin at Rosebank, the UK's largest known untapped oil field, which is located off the coast of Shetland in the North Atlantic.