1. do something that comes with a high degree of personal risk
he/she put his/her life on the line to save somebody
To put something on the line means to put something at risk. If you put something on the line means to behave in a way which means you may lose something such as reputation, money or a job. You can also use the expression to lay something on the line.
2. connect somebody or get somebody to speak on a phone
put your manager on the line, I wish to make a complaint
If you were on the phone you can say you are going to put somebody on the line, meaning you are going to add someone else to the call or put somebody on the phone.