A distribution T/F of 500kva (delta-Wye) connection, if its one phase of primary side (H.T delta side ) is faulty, is it better to use remaining two phase or not, if it use what failure will occur?

My understanding, for a 3-phase equipment, I1+I2+I3=0. So, if one phase is faulty (need to know what type of fault, if short to ground then confirm cannot use) then I1+I2+I3 shall no longer be zero. You will get an unbalance condition.

For 3-phase motor, if one phase is faulty, usually the motor will trip. We usually have such problem with the motor's contactor, 1 of the phase faulty hence the motor cannot run on 2 phase.

However, sometimes 3-phase loads are unbalance anyway, e.g. for lighting where each phase supply different loads.

However, because we are talking about a transformer, i am of the opinion that you should shut-down the transformer. Because when in unbalance condition, the other 2 phases shall definitely carry more Current, hence even if the winding & circuit still can cater the larger capacity current, it could probably shorten it's lifespan. But i don't know, probably other people have experience with such case.

which type of fault ?

If one of your phase in the delta primary HV side is faulty, do not use the transformer, you must not use it. until the faulty winding is repaired. If you try to use it you are creating a hazardous condition that may lead to a catastrophic failure or accident. Failure that may also involve upstream and downstream components of your syste.

Do not use it except as an emergency step. You do not know why one phase failed, so you don't if and when the next phase will fail, possibly with catastrophic results. Running two phases out of three with an auxiliary to make up the third phase almost guarantees that there will be an unbalance in the output voltage/current. This is due to the lack of a return path for the magnetic flux from the single phase transformer to join with the other two phases' flux, the result is poor regulation and possible overheating.

better not to use the unit or you will encounter worse scenarios later on, making the damage more severe... fix the fault on the HV side of your transformer before energizing it for operations...