The frog is not globally extinct. However, it is extinct from Louisiana. It does not live on the property in question, or in the state where the property is located. See Mississippi gopher frog could hop into St. Tammany.
As far as burning, a fifth circuit court of appeals decision in the matter states:
ninety percent of the property is currently covered with closed-canopy loblolly
pine plantations. These trees would have to be removed or burned and then
replaced with another tree variety to allow the establishment of the habitat
that the Service has concluded is necessary for the breeding and sustaining of
a dusky gopher frog population.
For more information, see Endangered and Threatened Wildlife
and Plants; Designation of Critical
Habitat for Dusky Gopher Frog
(Previously Mississippi Gopher Frog) Federal Register / Vol. 77, No. 113 / Tuesday, June 12, 2012 :
This rule designates critical habitat
for the dusky gopher frog.
Approximately 625 hectares (1,544
acres) are designated as critical habitat
in St. Tammany Parish, Louisiana.
Under the second prong of the Act’s
definition of critical habitat, we can
designate critical habitat in areas
outside the geographic area occupied by
the species at the time it is listed, upon
a determination that such areas are
essential for the conservation of the
Unit 1 encompasses 625 ha (1,544 ac)
on private lands managed for industrial
forestry in St. Tammany Parish,
Louisiana. This unit is located north
and south of State Hwy. 36,
approximately 3.1 km (1.9 mi) west of
State Hwy. 41 and the town of Hickory,
Louisiana. Unit 1 is not within the
geographic area occupied by the species
at the time of listing. It is currently
unoccupied; however, the last
observation of a dusky gopher frog in
Louisiana was in 1965 in one of the
ponds within this unit.
The land, according to court decisions thus far, has not been "taken" (within the meaning of the 5th amendment to the US constitution). The land has been designated as critical habitat for an animal species. The government "wants to" [meaning hopes that the owner will voluntarily consent to] burn the land to create good conditions (open canopy rather than closed canopy forest) for the species.
However, though the government is not trying to absolutely force the burning upon the land owner, as explained in Economic Analysis of Critical Habitat Designation for the Dusky Gopher Frog (a study commissioned by the government), the government can stop the owner from developing the land, by denying a section 404 permit, on the basis that the land is critical habit for the frog, which would cause $34 million dollars in lost value. The economic analysis report also says that if the land owner "works with" the government "to establish conservation areas for the gopher frog within the unit, resulting in 40 percent of the Unit being developed and 60 percent managed for gopher frog conservation and recovery", then there would only be a $20 million dollar loss.
Overall, the government is attempting to economically pressure, but not absolutely force, the owner to burn or otherwise remove trees to reestablish habitat suitable for a frog that currently lives only in the neighboring state.
For more information see this Supreme Court Petition for a Writ of Certiorari.
The Supreme Court has now (1/22/2018) decided to hear this case.
The Supreme Court unanimously reversed the lower court's decision, for two independent reasons:
Firstly, the Supreme Court held:
Only the “habitat” of the endangered species is eligible for designation as critical habitat.
and secondly the lower court did not adequately consider whether the decision to designate the land as critical habitat, despite economic consequences, was:
arbitrary, capricious, or an abuse of discretion