The liver can become depleted due to extended exertion which causes liver dump, not necesssarily at dawn, Once depleted then any further demand for energy must be met by other means, which is when (a) ketone adaption takes place so that the muscles and the nervous system become adapted to burning the basic acetyl compounds that go to direclty power the cells, such as CoA.
The body is adapted to survive this condition but it must raid other stores in the body. Firstly it takes the stored lipids held in the adipose tissue and rips them apart for the basic acetyls, and can normally do this for some time since these are the long term energy stores. But if demand continues then bgl gets very low and the brain gets foggy (hypo). In order to keep the essential services running, the body starts to shut down the peripheral muscle activity, in order to prioritize the glucose for the brain and vital organs. This is when gluconeogenesis begins, but it is an inefficient process and slow to supply new glucose from fats. It also scavenges proteins from other parts of the body such as muscle tissue. Now this is an extreme condition since the fat stores are normally kept fairly full with glucose and lipids. So we are talking true starvation conditions here where the body burns the books in the library to keep warm. This is why hypos are dangerous since coma can follow as the body shuts down,
One thing our body does well is run on fat instead of glucose, which is where the keto diet comes in and this allows us to survive for a while without food, It is also why the LCHF diet recommends a higher than normal fat intake so as to provide plenty of lipids to use up instead of scavenging muscle tissue. Its like having a dual fuel burner or a hybrid car,
So yes in answer to your question we do burn fat if there is low glucose and mother hubbards cupboard is bare. But gluconeogenesis is not a major player.