A current carrying wire produces magnetic field around it. We can find the direction by Fleming's Right hand rule. We know change in electric filed produces magnetic field and change on magnetic field produces electric field. It is mutual relationship. My question is that is there any condition such that current carrying loop or wire produces no magnetic field
Moving charge always produces a magnetic field. If you have a non-zero current then you have non-zero moving charge and a magnetic field will be produced.
You can achieve essentially no magnetic field though by using two wires right next to each other each carrying current in the opposite directions. As long as the wires are very close and the amount of current they carry is very close the magnetic fields they produce will nearly cancel. This is why a clamp meter can't measure current around two conductors carrying current in opposite directions.
If you making a solenoid you want to also wind all the loops in a certain direction. If you instead wound it in a short loop with an equal number of turns both clockwise and counter clockwise then there would be practically zero magnetic field. Brandon's examples apply because again we have opposing counter currents and symmetry so that the magnetic field cancels. A toroid also has about zero magnetic field outside of it, and almost all the magnetic flux is locked inside the toroid. An infinitely long solenoid (no such real thing exists) would also have zero magnetic flux outside but just a really long solenoid will suffice as an approximation. I am sure many other types of arrangements are possible.
Edit 1: I wanted to correct that an infinitely long solenoid has approximately zero magnetic field compared to the magnetic field inside. In truth the magnetic field never perfectly goes to zero. The same is also true of the toroid.
A current carrying loop arranged in the form of a twisted pair of cables with same diameter, will produce a nearly zero magnetic field.
The use of a coaxial cable will produce a still lower magnetic field.