According to standard BS 7821 part 4: A distribution x-former supplying non-sinusoidal load would end up with few percent of overloading with respect to same amount of sinusoidal load which is usually reflected in X-former sizing procedure by a k-factor, I suppose. However, taking up to 20% of spare capacity in distribution x-former sizing, this will hardly result in over temperature trip out unless in case of very marginal sizing design or poorly filtered non-sinusoidal loads e.g. variable frequency drives (VFDs).
In addition to above, it is useful for your further understanding to pay attention to the source of those harmonics.
There are basically two of them: 1) saturable magnetic devices such as generators, transformers, motors and other iron core thingies due to their hysteresis slope - they inherently generate harmonic currents of mostly 3rd and other triplen harmonics; 2) any current and voltage 'chopper' such as a regulated rectifier – these generate harmonics in accordance with number of their pulses or the pure algorithm outcome (such as in case of AFE rectification).
First group is not a big problem since triplen harmonics can be i) fully stopped from spreading in the system by pumping power through a delta-wye transformer, or ii) greatly reduced by introducing grounding Zig-Zag transformer at some critical points of the system, or iii) reduced by introducing filters – the worst possible choice since a power network has number of resonance nodes caused by distributed capacity and inductance. When a certain harmonic coincides with the node it excites the system as a resonance circuit. Consequences are usually overvoltage that leads to insulation degradation. The overvoltage level depends on the network load, type of load and other factors. And the resonance frequencies are not fixed, they have tendency to shift, affected by the same factors as the overvoltage level. To close this, triplen harmonics of course act as zero sequence component with all the implications associated with that.
But the second group is much more interesting. As an example, 6-pulse rectifier generates harmonics: K6=6xN±1, so that K6 = 5, 7, 11, 13, 17, 19 and so forth. Odd harmonics such as 5th and 11th etc. pass through the delta-windings and act as negative sequence component that creates negative torque in rotating machines, opposite to the normal torque. Thus, it's twice of the nominal frequency -> overheating due to eddy currents etc. -> overheating due to overload etc. And since the positive sequence impedance of rotating machines, such as AC motor, at standstill and the negative sequence reactance of the same machines at running speed are usually match, -> 1% of negative sequence voltage applied would cause 1% x LRC = 6-7% of negative sequence current to appear in the motor.
How to fight the issues associated with the second group? Stop buying last-century designs like 6-12-18 pulse drives and either go with higher pulse (but much more expensive) or use AFE that carries harmonics compensation mechanism inside the mathematical algorithm itself.