As mentioned above, the meter will still be fully functional, with a 1.6A fuse, rather than a 600mA fuse, but it's a bad idea, because you lose the protection.
One thing to note is that, in resistive loads, such as cables and the meter's current sense resistor, the power dissipation increases to the square of current. The power dissipation inside the meter and cables will increase by a factor of 1.62/0.62 = 7.11, when the current is increased from 0.6A to 1.6A. In reality, the short circuit current can be a thousand times those figures, causing a massive power dissipation, inside the meter, even with the 600mA fuse. If it's replaced with a 1.6A fuse and is overloaded, it may blow up. If the circuit has enough energy, this can cause serious injury or death to the person holding the meter.