How can we increase the range of a line sensor to make it work as a proximity sensor?
The line sensor you linked operates using an IR LED and phototransistor, and the specifications state that the range is adjustable from 10 to 50 mm.

To increase the range further, you would need to use a more powerful IR emitter and/or a more sensitive IR detector. The angle of view (sort of a misnomer when talking about infrared) would need to be considered as well. Some sensors employ plastic or glass lenses to focus and achieve greater ranges.

A higher-power emitter would probably require more power than the circuit is currently designed to supply, so you would also have to modify the current-limiting resistor and/or the supply voltage.

IR proximity sensors are designed to detect a certain threshold of reflected IR light. When a object that reflects IR is near, the phototransistor base current goes up. When the desired threshold is reached, the circuit produces "HIGH" (1) output. The line-following sensor is the same basic idea.

You might be able to use another sensor that is designed as a proximity sensor and already has the range you are looking for. It will require some experimentation and calibration, as the line or surface you are trying to detect will probably be more reflective than, say, a human.

Also keep in mind that longer range detection can incur additional problems, such as other sources of infrared light. If you're trying to detect a line outdoors, sunlight will raise the amount of detected IR across the board, and you'll have to change the detection threshold.