How does the potential of a bird increases after sitting on a wire due to which, it does not get electric current?

Current is the result of a difference in electric potential between two points.
If an entire object is at the same potential, then there's no reason for current
to flow anywhere over, under, around, or through it.  That's how an electric-repair
lineman goes up in an insulated bucket, connects himself and the bucket to the
line, and then proceeds to work on the live line without danger.

The difference in electrical potential across the entire bird is the difference
between his feet ... perhaps 1 to 10 inches along the length of the line that's
hundreds of yards or hundreds of miles long.  The potential difference  
between those two points a few inches apart is very small.

The objects are acted upon by only one force. gravity. mass times the acceleration of gravity (-9.81meters/second squared).