Hard to say initially, although I would check your buffer to see if the background is present within it.  I would also recommend running purified hen egg lysozyme as a control.  Lysozyme profiles are well studied and therefore will prove handy in your troubleshooting.  Good luck

I used 100mM Acetate buffer pH 5, without any salt. And is it necessary to centrifuge samples before the analysis? If yes, what speed and for how long?

Also, does SYPRO orange in any way interact with native BSA? couldn't find any literature on that.

SYPRO orange will very likely bind BSA, which nonspecifically binds hydrophobic molecules.  If your sample contains BSA, this is likely the source of the problem. 

I agree with Jeffrey, Sypro orange will bind to BSA.  I would definitely subject your sample to some sort of filtration such as amicon columns to remove any aggregates or high molecular weight aggregates.  Centrifugation at 10000xg should pellet large aggregates and inclusion bodies if they are present in your sample.

On more than one occasion I have asked visitor to my lab if their sample contains ONLY the protein of interest.  I then learn that the sample also contains BSA because the young investigator believes "you always must include BSA to prevent protein loss to the walls of the sample well".  I have since learned to specifically insist that BSA NOT be present in the sample.  I think that many molecular biologists who turn to biophysical measurements do not appreciate that BSA is a very active component with strong binding properties.

If I had to study the melting of BSA, what dye should I have used?

A great many dyes will bind to BSA.  For example, ANS is the classic probe with BSA.  But if your instrumentation permits it would be better to use the intrinsic tryptophan fluorescence or even light scattering so that you are not adding complications. 

Perhaps I should add that the problem with the use of many dyes in thermal studies is that unfolding of proteins may result in the appearance and subsequent disappearance of new hydrophobic regions which may bind dyes and then may lead to protein aggregation.  So fluorescence data on these systems must be interpreted with care.

Thanks a lot guys. 

An alternative to the classical DSF, which uses dyes to detect protein unfolding, would be the nanoDSF. This quite similar technology does not require any dye as it measures the intrinsic fluorescence of tryptophanes to monitor the unfolding. No dye, no problems with unspecific interactions!