Calculate the maximum current of zero ohm resistors
Unfortunately there is no such thing as a true zero ohm resistor (outside of superconductors). Zero ohm resistors available on the market (often called 0R link or jumper) will specify a maximum resistance. 

This datasheet is for a range of 0402 surface mount resistors, and specifies that 0R jumper resistors have a resistance of less than 0.05 ohms. It also specifies a maximum power dissipation of 1/16W. This implies that (worst case) the maximum current you can safely pass through the part is:
$$
\sqrt{ \frac {P_d}{R} } = I
$$

Which comes out to around 1.1A, for such a small part it sounds reasonable.

You must bear in mind however that for such low resistances, tracks/wires and solder blobs will have a big impact on this value.

If you need a lower resistance and/or more current capacity, parallel the resistors up.

EDIT: From a theoretical point of view, as the resistance decreases the power dissipation tends towards zero. So for a true zero ohm resistor, no power would be dissipated, allowing infinite current. If your interested in the theoretical side, then I suggest you read some Wikipedia articles on superconductors where the resistance truly is zero. Things get a bit weird.