In their literal senses, line of fire refers to the path or potential path of a projectile, and firing line refers to (1) a row of shooters directing fire at a target, or (2) a row of soldiers at the front of an assault.
Both phrases are also used figuratively. To be in the line of fire is to be in a situation in which one is either (1) likely to be attacked, or (2) in or near the paths of flying projectiles. And, logically, to be in the firing line is (1) to be among a group of people attacking a target, or (2) to be at the vanguard of something.
That’s the logical distinction, anyway. In practice, firing line very often takes the place of line of fire in the latter’s figurative uses, while line of fire is usually reserved for literal uses.
Although to be on the firing line is literally to be among those directing fire at a target, the phrase is often used figuratively in reference to those who are attacked or soon to be attacked—for example:
Facing the nine justices of the state supreme court, Mississippi Attorney General Jim Hood must have felt like he was on the firing line. [WJTV]
Not far away are tents housing representatives from Portugal, who fear they will be next in the firing line. [Guardian]
Retail jobs could be next in the firing line, after new figures showed retail spending fell over during Christmas. [News.com.au]
And though line of fire is sometimes used figuratively, it’s more often literal, and often comes up in reference to innocent bystanders who become casualties—for example:
Giunta put himself in the line of fire for two fellow soldiers during a 2007 firefight in Afghanistan. [The State]
Cautiously peering round doors the son pointed to the bullet holes in the windows, a hole in the roof from a mortar, and a deceptively easy line of fire. [Telegraph]
The Muslim Brotherhood, meanwhile, is trying to remain above the fray and out of the line of fire by making deals with the army. [New York Times]