How does the efficiency of a transformer change with it's load?
Increasing current causes increasing copper losses due to resistance.
 Resistive loss is determined by current^2 x resistance.   BUT,
 Power is determined by voltage x current.

So if you increase power from 100% to 110% the copper losses rise  by
 (110%/100%)^2 = 1.1^2 = 1.21 ie losses increase by 21% for 10% more power.

Copper resistive energy loss is turned directly into heat.
 A transformer will be designed to have a safe temperature rise at rated power in the worst case environmental conditions that it is guaranteed to work in.
 Add 20% more heat and things may get "interesting"/ Short term failures may occur.
 But, if not, inter-winding insulation will "cook" and perish, wire insulation may fail. 

Iron losses due to hysteresis will also increase. Brain offers that this will be linear with current but brain may be wrong.

Increased temperature may affect magnetic property of the core steel. IF core permeability drops even slightly then flux will drop and inductance will drop and current per applied volt will rise and copper loss will increase and temperature will increase and ... .
 Thermal runaway is not usually seen in domestic size transformers. Fortunately.

This page from Elliot Sound products notes:


Keeping a transformer as cool as possible is always a good idea. At elevated temperatures the life of the insulation is reduced, and the resistance also increases further because copper has a positive temperature coefficient of resistance.  
As the transformer gets hot, its resistance increases, increasing losses. This (naturally) leads to greater losses that cause the transformer to get hotter. There is a real risk of drastically reduced operational life (or even localised "hot-spot" thermal runaway) if any transformer is pushed too far - especially if there is inadequate (or blocked) cooling.
It is generally accepted that any transformer will have one part of the winding that (for a variety of reasons) is hotter than the rest.
It's also a rule of thumb that the life expectancy of insulation (amongst other things) is halved for every 10°C (some claim as low as 7°C).
When these two factors are combined, it is apparent that any transformer operated at a consistently high temperature will eventually fail due to insulation breakdown. The likelihood of this happening with a home system is small, but it's a constant risk for power distribution transformers.   
Despite all this, mains frequency iron cored transformers typically outlast the product they are powering, and even recycled transformers can easily outlast their second or third incarnation. Once a transformer is over 50 years old I suggest that the chassis be earthed, as the insulation can no longer be trusted at that age.  




Added 8 years on :-).

Not directly asked about but related and worth noting: Transformer manufacturers seek to minimise cost (of course) and using as little lamination material as reasonably possible is a target. The core is usually designed to operate at the knee of its flux BH curve where increasing amp-turns start to give increasingly less flux per amp-turn as the core is driven further into saturation.
Transformers designed for 60 Hz operation can use usefully less core material due to the increased impedance at higher frequency (Z = 2.Pi.f.l).
However, operating a transformer designed for 60 Hz in a 50 Hz environment can lead to very substantial excess heating.
While this is not normally encountered it does happen. Two examples:


People bringing equipment from eg the US to NZ not only need to adjust transformer tappings (if availaable) to accommodate the 100 VAC to 230 VAC change but also need to take account for the change from 60 Hz (USA) to 50 Hz (NZ)
I once had a custom 500 watt mains power transformer would in New Zealand for use in a test box in a Taiwanese factory. Their mains is 110VAC and hours is nominally 230 VAC. I specified two primary windings that could be connected in series or parallel to allow operation in either country. In the specification I did not mention Taiwan but I did tell the manufacturer that it was for use in Taiwan, ultimately. He took it on himself, without asking or telling me, to design it for 60 Hz operation. NZ uses 50 Hz. While a 50 Hz design would have worked well in Taiwan, the opposite was not as true as I would have liked. In NZ on test it ran VERY hot - it took me a wee while to realise why.