on the line 作为一个短语,其意为“岌岌可危;冒极大的危险”。如:
Their careers were on the line. 他们的职业生涯岌岌可危。
He wouldn't put his career on the line to help a friend. 他不会为了帮助朋友而让自己的事业受到威胁。
所以你说的 lay something on the line 可以理解为:让某事物受到威胁;让某事物处理危险之中。