I've read that you can use a transformer to step up or step down the voltage when the voltage is AC(i.e. the average value of voltage is 0). What would happen if the average value is not zero?
The average DC voltage will ramp the current up through the inductance of the transformer, according to dI/dt = V/L. This will happen relatively slowly.
At some modest current, well below the rated load current of the transformer, the core will saturate, reducing the inductance. The current will then rise much faster, until being eventually limited by either the DC resistance of the winding, or a breaker blowing.
If the average voltage is very small, you can get a quite subtle effect. Many years ago I built an 'easy to use' hifi system, whereby switching on the CD player for instance would cause the amplifier to switch on as well, and the pre-amp to select the correct input. I used a few silicon diodes in series to get sufficient voltage to drive an opto-isolator input, in series with each peripheral to detect whether they were drawing current. Being cheapskate, I used three diodes to develop sufficient voltage, but only one in reverse. This difference of two diodes drops caused sufficient mean DC into the transformers in my various peripherals that they audibly hummed, due to the magnetorestriction effect of being saturated. Balancing the diode drops silenced them.
Most likely half wave saturation. Saturation on 1/2 of the AC cycle, out of saturation on the other half. It is an issue associated with geomagnetically induced current (quasi- DC) flow during solar flares for the power system.
Here is a good read with a section on the effects on transformers that should help you.
The coil of a transformer typically has some resistance. There will be a DC current flowing through that resistance if there is some DC component.
If the current is large it could cause the transformer core to saturate and/or it cause excessive \$I^2R\$ heating in the transformer winding.