Burley pointed out that the prime minister promised no one would have to sell their homes for social care at the last election.
"I’m only doing what you normally do, but carry on," the health secretary said when the news presenter tried to ask a question.
"It looks like the direction which Sky News wants to go over the next few years is not one that’s a particularly good fit for me."
Tories accused of "wallowing in sleaze" as former cabinet minister spared immediate suspension.
"If they haven’t got vaccinated by now, then there is an issue with patient safety," the health secretary said.
Broadcaster and channel's chair has been largely absent since announcing a break two weeks into launch.
"I let myself and my viewers down," the presenter said.
After a launch week involving Ofcom complaints, technical blunders and embarrassing pranks, the former BBC News star has spoken out.
"A quick thought before I ask you one more question, because it does look like Mr Opik really does want to talk to you..."
The presenter was taken off air after attending a gathering for her 60th birthday when London was under Tier 2 restrictions.
The government is expected to win support for keeping hold of powers until October.
Education secretary said teachers were among public sector workers set to face "pay restraint".
The transport secretary said the number of people coming from the 33 "red list" countries was "less than 1,000 people a day".
The government issued guidance on New Year's Eve saying it was "reasonable" for two different vaccines to be given to the same patient if one type was unavailable.
SAGE has recommended all secondary schools remain closed in January.
Ministers and scientists have previously said they think the vaccine would become effective by spring.
Labour MP Siobhain McDonagh wasn't having any of it.
Beth Rigby and Inzamam Rashid will also remain off air for three months.
The Sky News broadcaster admitted breaking safety rules after celebrating her 60th birthday in London.