What voltage is put across the Tx+, Tx-, Rx+, and Rx- lines over ethernet?
This isn't practical. Since modern Ethernet has a code, there are transitions even when there are no packets. The LEDs you see on networking equipment are driven by the PHY, not straight off the cable.

Even given that, the transitions are way too fast to see it go off an on with tranisions. The light would be apparently always on, and not blinking.

If you wish to wire up a signal, the fast Ethernet (100BASE-TX) voltages are +1 V, 0 V, and -1 V. I believe it is ~100 ohms. 1 V/100 ohms = 10 mA. If your LED is in parallel, you need it to not effect this too much.