The tint slider takes care of a couple things. First off, from a color perception standpoint, there are two major axes that the cones of our eyes base color perception on: blue/yellow and magenta/green. There are some specific nuances related to these axes, however the most important is that they represent opposite colors that the human eye can not see simultaneously at any given single point (i.e. you can't see blue and yellow at the same location point...the eye oscillates between the two....however you could see blue and or green.) Most color calculations that involve human perception are done in Lab* space, which is fundamentally based on those two axes of color.
When it comes to the blue/yellow axis, this happens to be the primary axis along with white point falls. White point is a critical component of color balance, and is generally why we make color temperature adjustments far more frequently than we do tint adjustments. This is partly due to the kinds of illumination we normally use to light photographic scenes as well. Sunlight, incandescent lighting, and a considerable amount of photographic artificial lighting is achieved via black-body emission lighting...matter heated to the point of emitting light. Such lighting usually emits light along the blue/yellow axis, rather than the green/magenta axis.
Another form of lighting, gas-emission lighting, usually emits light more in line with the green/magenta axis. Gas-emission lighting is usually found in fluorescent lights, neon lighting, etc. Scenes photographed under gas-emission lighting will usually take on a greenish or slightly magenta-tinged appearance (at the best of times), or may end up with a more aberrant appearance. The Tint slider in Lightroom is a way to correct for "off-axis" color balance issues often caused by gas-emission lighting or any form of non-standard (blackbody) lighting.
Generally speaking, color balance corrections will be done primarily with the Color Temperature slider, with minor adjustments to the Tint slider. Alternatively, the color temperature and tint sliders can be used to accomplish artistic effect, adding a strong color cast to your photos when one is absent. You may have a scene that is slightly purple-tinged, and one can enhance the artistic appearance of that purple tinge by making a larger-than-normal tint adjustment towards magenta.