Long term effects of operating in Saturation mode of transformer
When the magnetizing curve goes above the knee point of saturation, then the inductance is reduced to the air core inductor. Therefore with absence of reactance the impedance of the transforemer becomes only the resistance of the winding. In such scenario for short time the transformer acts like short circuited. It is now regarding the mains impedance how much current it will flow from mains. In domestic network the xformer will heat up without any big noticable fact, while in industrial network it will blow the fuse, since the mains impedance is lower. 
The xformer in saturation condition will excessively heat, this is the effect.