Arduino Uno: Pullup Resistor
Each pin has its own, separate, pullup resistor. 

I don't believe you can damage it by drawing current from the port, because the pullup resistor is only active when the port is in high impedance (hi-Z, input) mode. 

The pullup resistor can be damaged by overvoltage, static discharge, and the like. 

The Uno has 20 kOhm pullup resistors according to this tutorial. If you feed it a 0V signal while the Atmega is running at 5V, the current will be a quarter of a milliamp, which is miniscule.