The maximum current rating considers both current carrying capacity, and arcing during disconnects, which is also why you have separate AC and DC ratings (arcing is less of a problem with AC, as the arc would be interrupted during the zero crossing).
Short pulses that exceed the DC, but not the AC rating should be fine, but if you turn off at an inopportune time, that may shorten the lifespan a bit.
Exceeding the AC rating means warming up the conductors, especially at the point where the moving parts touch as that is going to be the narrowest point of the connection. If you weld the connectors together, the relay may fail closed.
If this is about inrush current after turning on, you can probably also add an inductor in series after the relay to give it a slow start, but keep in mind the inductor needs a flyback diode (so this can be used for DC only).