Should a UPS transformer heat up on no load?
The symptoms you describe are consistent with it working properly. 

Note that I'm not saying it's not broken or that it's working properly, that's impossible to diagnose at this distance with the limited data you've given. However, what you've told us is not sufficient of itself to be alarmed.

There are several sources of heat dissipation in a transformer. There are core losses, which are constant when the transformer is operating, whether at no load or full load. These are due to the flux swings in the core. Then there are copper losses, due to two components of current in the coils. There are magnetising current losses, which are very small in reasonable size iron transformers, but can be quite significant in small iron transformers, ferrites, and microwave oven transformers. Finally there are load current losses, which vary as the square of the output current. As these are in quadrature to the magnetising current, they generally dominate the magnetising current losses at full load.

So at no load, there are two sources of heating inside a normal transformer.

The fact that the fan doesn't come on at no load suggests the UPS designer has done his work, and sized the transformer reasonably. A warm transformer is not a cause for concern, especially when it's operating without the full cooling intended for full power operation.