To become shiny or glossy (often due to oil or sweat). A colloquial verb used for faces or objects that shine.
暑くて額に汗をかき、すぐにテカってしまった。
It was hot and I sweated on my forehead, which quickly became shiny.
靴を磨くとつやが出てテカる。
When you polish shoes they gain a gloss and shine.
メイクが崩れて鼻がテカってきたからティッシュで押さえた。
My makeup broke down and my nose started to shine, so I blotted it with a tissue.