In order to improve the transport properties and magnetic properties of Strontium iridate, which elements should be replaced at strontium position or iridium position?
Depends on what you mean by "improving" the transport properties. There are many different compounds in the perovskite family that will introduce ferromagnetic or ferroelectric order:
Niobium (strontium niobate, strontium barium niobate, barium niobate) for ferromagnetics
Lithium (lithium niobate, lithium iridate, lithium titanate) for ferroelectrics
Titanium (strontium titanate, barium titanate) for ferroelectrics
Tanuj is right, in that it depends on what you mean by improving the transport properties. In common with the isostructural cuprates, there are two main methods of modifying the magnetic and transport properties: electron doping and hole doping. These have the effect of introducing additional charge carriers that perturb the electronic and magnetic ground state. The prototypical example are the hole doped cuprates such as YBa2Cu3O7-x, which is superconducting for a given value of x.
Now return to the case of Sr2IrO4 (I assume this is the system you mean, there is an entire Ruddlesden-Popper series of them: Sr3Ir2O7, Sr4Ir3O10,.... SrIrO3). If you want to hole dope your material, then this can be done either by annealing so you have Sr2IrO4-x, substitution on the Ir site with Rh for example (see Calder, PRB 92, 165128 (2015)), or substitution of the Sr site by K.
On the other hand, if you want to electron dope your material, then doping with La seems to be the best bet: (Ge, PRB 84, 100402(R) (2011)). The issue with La doping is that it is very difficult, if not impossible, to introduce more than around 8-10% La into the structure.