1) Sew it up! This works better on mesh tights (like most seamed tights) than on microfiber tights as sewing those can sometimes make a hole worse. Grab thread that matches your tights and, if you're dealing with a more circular rip, stitch around the hole like a clock (working from 12 to 6, then 1 to 7, and so on). If the hole is a more oblong shape you may need to stitch from left to right, instead.
2) Apply clear nail polish around the edges to keep the hole from getting worse. This is best if you don't need to physically close the hole (like if it's covered by your costume), and if you've got time to let the nail polish dry before you have to be on stage.
3) If you're really in a pinch—or to be extra sure a hole won't run—grab your hairsprayand spray the edges of the tear. The hairspray will dry slightly faster than nail polish, so this can be great if you're in the middle of a performance when a rip happens and you need a quick fix to keep the situation from escalating.
*You have an extra pair of tights in your bag for this exact situation, right? Right!?