No true for a short distance, they would almost draw no current at all.
When you connect long wires to the serial RX and TX pins, many bad things can happen.
Voltage peaks could damage the Mega board; The communication could be too slow; and it is indeed possible that the TX causes current peaks. The RX and TX signals are they are, are not suitable for long wires.
There are many ways to use long wires. Line drivers, chips, filters, circuits, protocols, and so on.