Industrial transformers should have temperature ratings on the label. They are designed for maximum working temperatures. Probably not the case for transformers in consumer gear like domestic UPS, etc.
If the transformer is getting so hot that it is emitting odors, It seems likely that you are overloading it. At best, it will likely reduce its expected life-cycle, and at worst, it may be a fire hazard.
Many people connect things to UPS that don't really need it. The only things that should be operating on UPS are the things necessary for orderly shutdown in case of power failure. That doesn't include things like printers, scanners, lamps, etc.