Reasons for Damage/Burning of Distribution Transformers

The simple answer is no or improper maintenance. Go through manufacturer's literature.

I expect technical deliberation from the learned members of this forum instead of astraying from the thread.

Expectations can be very dangerous things.

Here, the quality of the answers can be taken as a reflection of the quality of your question. If you ask a vague, general question such as in your OP, the responses you get will reflect that vagueness and generality. If, however, you ask a more specific question that shows that you have already done a bit of research, you will ge a ton of helpful answers.

You need to provide more information about failures.

Failures could be anything from someone shooting a bullet through the transformer to poor transformer design to overloading to old age.

Apart from general reasons, like overloading, unbalancing, improper earthing,penetration of moisture, etc. the present day load pattern like use of CFLs and other electronic loads, harmonics are produced travelling towards the transformers. The third harmonics has a significant contribution towards distortion of wave shape; that does not remain sinosidal rather distorted. It has significant impact on the core performance that needs to be studied. Further, the the use of CFLs would reduce the Power factor of the system; the affects thereof also need to be studied. Hoping my friends would comment accordingly keeping in view my submission.

Inadequate circuit protection arrangements are the principal reason, Guvnor, followed by a regime of inadequate maintenance.

there are no significant reasons bcoz everything can be significant, even small stuff could buy you a problem. here i give some reason :
first, unbalance load. in my country a distribution transformer is three phase transformer but each phase is loaded individually due to household voltage requirement wich we know mostly is single phase load. if for some reason the load in each phase is unbalance above it's service factor, it can cause damage
second, dirty post insulator. if a transformer is installed in dusty environment, the dust could cover the insulator and cause severe short circuit
third, dirty oil transformer (certain transformer). every oil transformer hast it's own lifetime due to it's liquid material. if a transformer is energized, the oil prevents it from short circuit between each phase by dissipating the electron leaps. this process creates small bulb/foam which will accumulate (some transformers equipped with silica gel to absorb this bulb/foam, it's color will change from original blue/violet to light blue from time to time as it absorbs the foam). and i guess you know what will happen if the foam accumulates above it's reasonable level. the foam contains oxygen which will react with oil transformer, creates sulfur film covering the conductive rod in each phase which means shorten the gap between each phase and bang....!!! there you go short circuit
that's all i know, hopefully it's helpfull