stay alert, pay attention, keep your senses sharp

Yes, as Neal and froopy said This is a term usually used in soccer, for example if referee gives somebody a yellow card then the player should be on his toes for the rest of the game. That means he has to watch out carefully not to pick another yellow card.

Neal is right, it means to stay alert and ready for action. I think this comes from playing sports. If you are standing causally, leaning back, with your weight on your heels, you can't react very fast if someone throws you the ball. If you are "on your toes", meaning your weight is slightly forward, and you're standing on the balls of your feet (the bump of the bottom of the foot next to the toes, that makes a deep impression when walking in the sand), then you are more agile and able to react to something. If you tell someone "be on your toes" it tells them to pay attention and be on the lookout for something. You would tell this to a goalie in a soccer game, for example.

我觉得这句话应改为： Keep yourself on your toes. 这样比较符合语法规则，但我也不知道俚语该怎么说。

不好意思，少写了复数。Keep sb on his/her toes.