I'm just trying to answer your question, but I don't think you're exactly clear on what you want to know. It's ok, no need to get defensive about it.
Whether or not 'one end of the wire gets heavier' depends on what your doing. If you are using the wire to power an LED from a battery, then no, because electrons are removed from one end and placed into the other at equal rates. Charge and mass within the wire are both globally and locally conserved.
If you do something where charge/mass isn't conserved such as removing electrons from one side (i.e. by rubbing a fork on a carpet) or by using an electric field to 'tilt' the electrons to one side, you can create a (very small) mass imbalance. This activities are not usually considered to be useful electrical current.
Whether or not 'one end of the wire gets heavier' depends on what your doing. If you are using the wire to power an LED from a battery, then no, because electrons are removed from one end and placed into the other at equal rates. Charge and mass within the wire are both globally and locally conserved.
If you do something where charge/mass isn't conserved such as removing electrons from one side (i.e. by rubbing a fork on a carpet) or by using an electric field to 'tilt' the electrons to one side, you can create a (very small) mass imbalance. This activities are not usually considered to be useful electrical current.