Burning fat transferred from one part of the body to another
Would the body use this transferred fat for as energy, or would it
  ignore it as it wasn't stored there in the first place?


It would use it as energy source, there is no such thing as "ignored tissue". Fatty acid mobilization is regulated by epinephrine and insulin. These are hormones which are ofc. carried by blood, so every white adipose (fat) tissue is affected by them.




Figure 1 - FFA and glycerol mobilization - source
2003 - Fatty acid mobilization from adipose
tissue during exercise



  Does the body differentiate fat cells stored in different parts of the body?


This is obvious, just check breasts and butts of girls. ;-) (Those are subcutaneous fat depos. They size is probably regulated by estrogen.)

There are other things involved here, we can talk about subcutaneous (fat below the skin), intramuscular (fat in the muscles) and visceral (abdominal fat) white adipose tissue (there are white, brown and beige adipose tissues). The visceral fat is associated with metabolic diseases, obesity etc... 


1995 - Pathophysiology and Pathogenesis of Visceral Fat Obesity
2012 - Lower Thigh Subcutaneous and Higher Visceral Abdominal Adipose Tissue Content Both Contribute to Insulin Resistance
2013 - Genomic and epigenomic regulation of adipose tissue inflammation in obesity


The regulation of body fat distribution and amount is complicated and regulated differently by subcutaneous, intramuscular and visceral fat. 


  The total transmissible variance ranged from about 40 percent for the
  amount of subcutaneous fat to 60 percent for the pattern of
  subcutaneous fat distribution. Biological inheritance accounted for
  only 5 percent of the variance for subcutaneous fat and the body mass
  index, but 20 to 30 percent for the percentage of body fat, fat mass,
  fat-free mass, and fat distribution. These data suggest that the
  amount of internal fat is influenced by heredity more than the amount
  of subcutaneous fat. Furthermore, we consistently found that
  nongenetic influences are quite important in determining the amount
  and distribution of body fat in the population.



1988 - Inheritance of the amount and distribution of human body fat 



  These data suggest that adiponectin concentrations are determined by
  intra-abdominal fat mass, with additional independent effects of age
  and sex. Adiponectin could link intra-abdominal fat with insulin
  resistance and an atherogenic lipoprotein profile.



2003 - Relationship of adiponectin to body fat distribution, insulin sensitivity and plasma lipoproteins: evidence for independent roles of age and sex



  The FFA storage pathway, which had remained undetected in
  postabsorptive humans until recently, can have considerable,
  long-term, and sex-specific effects on body fat distribution. It can
  also offer a way of protecting the body from excessive circulating FFA
  in obesity.



2011 - Storage of Circulating Free Fatty Acid in Adipose Tissue of Postabsorptive Humans
Quantitative Measures and Implications for Body Fat Distribution


Different colors (white, brows, beige) and orgin (subcutaneous, intramuscular, visceral) originate from different progenitor cells.


  These results reveal a major ontogenetic difference between visceral
  and subcutaneous WAT, and pinpoint the lateral plate mesoderm as a
  major source of visceral WAT. They also support the notion that
  visceral WAT progenitors are heterogeneous, and suggest that
  mesothelium is a source of adipocytes.



2013 - Visceral and subcutaneous fat have different origins and evidence supports a mesothelial source



  Brown and white adipocytes have been shown to derive from different
  progenitors. In this study we sought to clarify the molecular
  differences between human brown and white adipocyte progenitors cells.



2014 - Comparative gene array analysis of progenitor cells from human paired deep neck and subcutaneous adipose tissue


The regulation of fat distribution is under research, and not fully understood yet. 


  Fat distribution differs in men and women, but in both sexes, a
  predominantly gluteal-femoral compared with abdominal (central) fat
  distribution is associated with lower metabolic risk. Differences in
  cellular characteristics and metabolic functions of these depots have
  been described, but the molecular mechanisms involved are not
  understood.



2012 - Distinct Developmental Signatures of Human Abdominal and Gluteal Subcutaneous Adipose Tissue Depots