Controlled burns are not massive fires.
Several commenters have already mentioned this, but I thought it was worth fleshing out in an answer.
Controlled burns typically cover a few hectars to a couple thousand hectares. A thousand hectares is about 10 sq km (4 sq miles). A dragon's territory is likely to be 40-1000 sq km. The minimum size would overlap with the largest of controlled burns, but the maximum size would be 100 times larger.
Assuming the controlled burn covers the dragon's whole territory, controlled burns are designed to clear underbrush, and do not greatly impact local fauna, other than to change the available food sources. For predators, the controlled burn may even increase their available food sources long term:
Prescribed fire has an indirect, positive effect on large carnivore populations due to the high quality ungulate habitat it creates. (Source: Effects of Prescribed Fire on Wildlife and Wildlife Habitat in Selected Ecosystems of North America)
Even uncontrolled wildfires have limited immediate population effects,
Despite the perception by the general public that wildland fire is devastating to animals, fires generally kill and injure a relatively small proportion of animal populations. ... Animals with limited mobility livingabove ground appear to be most vulnerable to fire-caused injury and mortality, but occasionally even large mammals are killed by fire. The large fires of 1988 in the Greater Yellowstone Area killed about 1 percent of the area’s elk population (Singer andSchullery 1989)
(Source: Wildland Fire in Ecosystems)
After the controlled burn, much of the tree cover will still be intact, so finding the dragon will still present a challenge. I couldn't find a picture immediately after a controlled burn, but in the following image, you can see the large trees are still present.
So what would work? Defoliation
Burning a big chunk of forest to drive the dragon out of hiding sounds more like a "salt the earth" defoliation strategy, similar to using Agent Orange during the Vietnam War to "was to defoliate rural/forested land, depriving guerrillas of food and concealment and clearing sensitive areas such as around base perimeters" (Wikipedia).
These strategies were disastrous for locals. The military explicitly aimed to "destroy the ability of peasants to support themselves in the countryside" in the short term, and the herbicides had long-term health effects for as many as 3 million Vietnamese and 40,000 US Veterans (Wikipedia). For your fire-based defoliation, health effects could easily result from smoke inhalation instead of chemical exposure.
Large fires such as those that occurred in Australia and the Pacific Northwest this year are more similar to these military strategies than they are to controlled burns, in both scale (3,100,000 hectares sprayed with defoliants vs 400,000 hectares burned in Oregon - 18,000,000 million burned in Australia), and in ecological impact (covered extensively in @Ash's answer).
The military does not typically have the ecological health of an area in mind, and the local community is likely right to worry about the impact of military intervention.