If we analyze what a transformer is, it is a length of signal-conducting wire (s) that are wrapped around a permeable core. As a purely passive device, it has less to go wrong than with an active amplifier. The possible points of failure due to too-exuberant burn-in would be the core, and the wire.
Unlike a phono cartridge, a stepup transformer does not need to maintain a permanent magnetic field, therefore temporary saturation should not cause problems. However, I wonder how effective saturation will be in the burn-in process, since saturation will remove signal amplitude variations that the transformer core "sees".
If the current pushed through the signal wire is too high for its diameter, the wire will melt. But the wire diameters used in a transformer are certainly bigger than in a phono cartridge. It isn't so likely that a line-level signal will damage the wire.
If you use the output of a power amp to drive the transformer, or connect the transformer to the AC mains, certainly the signal wires could be damaged.
As long as you make sure that the current level used in the break-in process doesn't exceed the current-handling abilities of the transformer's internal signal wire (which will be affected by load / termination as well as signal level), I doubt if anything nasty will occur.
BTW, a transformer is a bi-directional device. Normally feeding a signal into the primary windings (normal input) will cause a stepped-up signal of equivalent power (higher voltage, lower-current) to appear at the secondary windings (normal output). However, the transformer can also be connected back-to-front and used as a step-down transformer - if you feed a signal into the secondary windings, you will get a stepped-down signal output from the transformer primaries (output signal voltage will be the inverse of the transformer's stepup ratio).
Needless to say, the above can be utilized when burning in an MC transformer (in which case, don't forget to load the primary windings).
hth, jonathan carr