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Sr₂IrO₄ is a spin-orbit-coupled insulator with a Neel temperature T_N=240 K. Our investigation of structural, magnetic, transport and thermal properties of Sr₂Ir_1-xFe_xO₄ with 0<x<0.34 reveals that substituting 3d Fe⁴⁺ (3d⁴) ions for 5d Ir⁴⁺ (5d⁵) ions (or hole doping) in Sr₂IrO₄ retains the native crystal structure but alters the physical properties in an unexpected manner. In particular, for light Fe doping (0 ≤x ≤ 0.14), the Neel temperature remains essentially unchanged but the electrical resistivity drops by five orders of magnitude, exhibiting a metallic behavior above 100 K. However, when the Fe doping level approaches 34% or x ~ 0.34 the magnetic order is eventually suppressed and the resistivity rises slightly but the insulating state is never recovered. Such unparalleled changes in magnetic and transport properties observed here indicate an unconventional correlation between spin and charge, raising questions of the role the antiferromagnetism plays in the iridates. The results will be discussed along with comparison drawn with Sr₂Ir_1-xCo_xO₄, in which both antiferromagnetic and insulating states readily diminish upon slight Co doping.