Just to add a bit of extra info to Dave Tweed's answer: -
What happens (besides heating) if the transformer's current rating is
It gets warmer under heavier load conditions and this is due to copper losses (\$I^2 R\$) in the windings. But, a side effect of this is that it might also increase the cores magnetic permeability and this means a higher flux density and slightly higher core losses.
Nothing so sinister really until the transformer is powered off and then back on again. Inrush current will have increased (especially if re-powered under no-load conditions) and this could blow a fuse.
When this happens, folk naturally think - why should it blow a fuse on repowering with no-load just because it's warmer. It's because the peak flux density seen in the the first AC cycle can be twice that at all other times: -
Why should the load being disconnected make this worse? If fully loaded the volt drops in primary due to the load current slightly decrease the magnetizing current and can partially avoid the full effects of saturation.