Mike Down, of accountancy firm Baker Tilly, said the HM Revenue & Customs (HMRC) should be checking tax returns to see whether individuals had already explained their circumstances.
He told the Daily Telegraph that The Revenue is "adopting a computer says yes approach, rather than simply checking the tax returns".
One case involved an elderly widow whose effective tax rate was low because she was giving more than half her income to charity.
More top news
The prime minister's announcement came as international leaders gather in Paris on the second anniversary of the climate change plan.
Prime Minister Theresa May said in October the Government will shift towards an organ donation system in England which presumes consent.
Gove's comments came as a draft bill enshrining animal sentience into UK law was published.