The 27-year-old made four appearances at the Rugby World Cup but will miss the forthcoming Six Nations.
The news comes as a further blow to the Dragons who only last week were forced to announce the retirement of former captain and hooker Tom Willis.
"Lloyd has been advised to withdraw from rugby for at least 3 months in order to fully rehab his neck," said Dragons Director of Rugby Robert Beale.
"We are obviously disappointed for Lloyd, particularly with the Six Nations coming up, but Lloyd's safety and welfare has to be paramount."
The retirement of Tom Willis and Burns' injury leaves the region with the trio of Steve Jones, Rhys Buckley and Sam Parry covering the position.