Both ways of understanding fire are possible. What we will discover is by taking both paths we end up in the same destination more enriched than were we to chose one path over the other. This is the beauty of the Greek language and God's word.
Edersheim's translation captures a practice every Jewish listener of Jesus' day would have understood. Before a burnt sacrifice was offered to God, both the body of the animal and the wood fueling the fire to consume the animal had to be first salted. From this vantage point, then, Jesus is using "fire" to represent a sacrifice--more specifically, an acceptable sacrifice.
In his letter to the Romans, Saint Paul defines what it means to be an acceptable sacrifice:
"I urge you therefore, brothers and sisters, by the mercy of God to present your bodies as a living, sacrifice, holy and pleasing to God--your acceptable worship; and don't conform to this age, but be transformed by the renewing of the mind so that you prove what is the good, pleasing, and perfect will of God." [Rom. 12:1-2]
We are to be living sacrifices to God, where are acceptable worship--service--comes through a renewing of our minds--that is, a fundamental change in the way we view the world. This change of mind starts with a purification of ourselves in purpose, motives, and thought--the whole person; this so we can effectively love others by seeking their spiritual and physical good for their sake and God's, in even the simplest ways; and this is done without harboring prejudices or divisions--something we easily do to elevate ourselves over others. In short, we become the least of all and the servant of all. Only when we become such people are we salted, and we are an acceptable sacrifice for God. Because only in this salted condition does God's love flow in justice that is true peace.
The more common translation sees fire as the means by which we are salted. The listener would have easily understood this allusion because, for example, precious metals are purified by melting them in a fire, so the impurities (dross) can be easily poured off. From this perspective, then, Jesus is teaching us that to become salted--to be salt--we must be purified as by fire.
In the first century, salt was a precious and therefore expensive commodity. Salt was used to preserve foods, and was seen as a nutrient. We become like salt to the world only when we listen to and yield to the Holy Spirit working in each of us who stand in Christ in God's kingdom.
The Holy Spirit will move us away from all those things causing us to sin (disobey God)--those attitudes and ideas we use to promote ourselves at the expense of others, and the selfish-ambition keeping us from truly loving others in even the simplest ways. In short, the Holy Spirit is purifying us with the fire of grace so we will become the least of all and the servant of all, which is to be real salt to this world.
This purification by the Holy Spirit will, like fire, be hot and painful--that is, it will necessarily entail suffering. The reason is everything the Spirit is transforming us into is in direct contradiction of how the world thinks and operates. The world strives for the single goal of being the greatest by wielding the most power. For this reason, we who strive for Christ's principle of greatness will experience resistance both from within ourselves and from outside ourselves; indeed, we will be hated, and such hate will incur suffering.
Even though the present world doesn't recognize it, and even violently opposes the idea, the world will not find peace outside the kingdom of God. We who stand in God's kingdom in Christ are His salt to this dark and confused world. We nourish and bring preservation by being living sacrifices, which is to become the least of all and the servant of all. If we cave in and hold onto the world's principles (i.e., conform to this age) in order to save ourselves, we lose our saltiness. This is what Jesus means by "if salt becomes unsalty, by what way will you season it?" And Jesus tells us elsewhere such vapid salt will be thrown out to be trampled under people's feet (see Matt. 5:13).
Therefore, my dear readers, let's stand in God's kingdom in Christ by submitting ourselves wholly to His grace through the power of His Holy Spirit working within us, and become the least of all and the servant of all. Only then shall we be at peace with each other.