The main problem is that the 360 was never designed to work at 1080 anything in the first place, the system was optimised around 720p and the thermal output that running at that resolution produces, running at 1080 makes the GPU run significantly harder and pushes the system outside of the thermal envelope it was originally designed for.
That doesn't mean that the GPU isn't capable of 1080 output, it means that the system would need to be thermally redesigned to cope with the extra heat, something along the lines of:
Redesigning the case & cooling system to remove any thermal hot-spots
Upgrading some of the components to more temperature resistant versions
It's not like every 360 will suddenly fail when pushed to 1080, but you'll end up with higher failure numbers depending on usage patterns.
I'm sure you'll get less problems with newer versions, especially as new hardware revisions come along and components go through die-shrinks