In the Hebrew Bible, salt is both a disinfectant and preservative, but if the salt loses its integrity (or its "flavor" to preserve) the result is disintegration. When Jesus talked about salt "trampled under feet," he was referring to this latter connotation of disintegration found in the Hebrew Bible. So when salt maintains its integrity (or its "flavor" to preserve), the effects are long-lasting (permanent), but when salt loses its integrity, the result is disintegration. The following paragraphs will explain.
The Hebrew noun for salt is מֶלַח, and the denominative verb (that is, the verb that is derived from this noun) is מָלַח, which means "to salt." There is also a second meaning to this verb, which is to disperse in fragments and therefore there is the idea of disintegration. The idea here could be of pulverizing a block of salt, although the idea of pulverizing by this verb is not confined to salt. For example, the Hebrew word for rags is מְלָחִים, which is a cognate of the same root מ-ל-ח. That is, rags are made by "ripping apart" or disintegrating a piece of cloth. So the meaning in not restricted to literal salt. The following example will illustrate.
The disintegration meaning of the verb מָלַח is found only once in the Hebrew Bible in Isaiah 51:6, and the context refers to the sky. The verb here occurs in the Niphal (perfect), which is the passive voice. The italicized bold text highlights the verb.
Isaiah 51:6 (NASB)
6 Lift up your eyes to the sky,
Then look to the earth beneath;
For the sky will vanish like smoke,
And the earth will wear out like a garment
And its inhabitants will die in like manner;
But My salvation will be forever,
And My righteousness will not wane.
So this verse in Isaiah literally says in the Hebrew that the atmosphere "will be dispersed into fragments," that is, it will lose its integrity and vanish because the simile at hand is smoke, which dissipates and therefore disintegrates. The parallel passages which corroborate the meaning of disintegration are Isaiah 34:4 and Psalm 102:26. (Isaiah 51:6 is also mentioned in the Christian New Testament in Hebrews 1:11, where the Greek verb is ἀπόλλυμι, which means "to perish.") So this verse here in Isaiah 51:6 is the only example in the Hebrew Bible of the negative connotation of מָלַח, which is the idea of disintegration. The remaining examples below will speak to the other meaning of the verb (and noun), where the integrity of "salting" is maintained.
When the Lord "salted" Sodom and Gomorrah with fire (cf. Mark 9:49), the result was the Sea of Salt (or Dead Sea). The salt acted as the antiseptic, and left the area in a permanent state of sterility. The sterilization of salt therefore leaves the land and water in a permanent state of desolation (Deut 29:22-23; Judg 9:45; Job 39:6; Ps 107:34; Jer 17:6; Ez 47:11; Zeph 2:9). Another example, but on a more positive note, was Elisha, who used one small jar of salt to make river water potable (2 Ki 2:20-21). In a similar vein, small quantities of salt were used with water in the ancient world to make saline solution, which was an antiseptic to disinfect newborn babies (Ezek 16:4). So salt is an antiseptic or cleaning agent, but when used in large quantities will leave both land and water in a permanent state of desolation. Thus salt here maintains its "flavor" to preserve whether for good or for evil.
Another example of purification was the salt used to sanctify offerings made to the Lord, and therefore the imperishability of covenant with the Lord. In this regard, the Brown-Driver-Briggs-Gesenius Hebrew English Lexicon (Peabody: Hendrickson Publishers, 2007) makes several references to the 19th Century scholar August Dillmann and to his edited work, Die Bücher Exodus und Leviticus (Leipzig: Verlag von S. Hirzel, 1880), which comments on the significance and meaning of "salt in covenant." Dillman writes on page 405 of his commentary of the salt mentioned in Leviticus 2:13 that
One held salt especially high, which had purifying and sustaining powers to make food tasty and imperishable; it was σύμβολον φιλίας, and was presented to guests before the best of foods (Eustath. ad Iliad 1, 449); it also functioned for the Orientals as a symbol and pledge of hospitality (Herbelot or. Bibl. Il. 773) and was a sign of covenant. When the Arabs affirmed a covenant, they placed salt on the blades of swords, and then placed the salt in their mouths; by which they became blood relatives and were faithful to one another even in mortal danger (Ritter Erdk. XIV. 960). A covenant of salt is, therefore, a covenant held to be inviolable and of permanent duration (Nu 18:19, 2 Chr 13:5). Yahweh and Israel had eaten salt with one other at the establishment of the theocratic covenant. This was always expected to continue in the service of the sacrifices, as the covenant itself was to last forever.
So Dillmann saw not only the sanctifying power of salt, but also its power of preservation. Thus covenant (whether with men or with God) was something pure and of long-lasting value. Dillman cited the Davidic Covenant (2 Chr 13:5) notwithstanding that "salt" is never mentioned per se in 2 Sam 7:10-17, where the Lord gives the covenant blessing to David. The "Covenant of Salt" of David is therefore pure (sanctified) and of long-lasting (eternal) value. The same is true of the Mosaic Covenant, where incense (Ex 30:35), grain offerings (Lev 2:13), and even meat offerings (Ezek 43:24) were to be salted. Even the Levites, to whom the Lord apportioned all of the holy "gifts" offered to God by the Israelites, were to be part of the Covenant of Salt (Num 18:19). That is, their apportionment of "gifts" was to be pure (sanctified) and was to be long-lasting (forever).
Thus when we come to the New Testament, we see Jesus using salt within the context of its use in the Hebrew Bible. That is, disciples are "salt," and so they are sanctified and therefore function as preserving agents. For example, we read the following in the Christian New Testament -
Ephesians 5:11 (NASB)
11Do not participate in the unfruitful deeds of darkness, but instead even expose them.
The believer is "salt," and the idea is the moral preservation of the society. Paul later tells believers that their speech "be seasoned with salt" (Col 4:6), and the meaning there is not only purity of speech (Eph 4:29), but that the speech have "gravitas" such as the giving of thanks (Eph 5:3-4).
So when Jesus is talking about salt losing its flavor and then is "trampled under under foot," he is alluding to the secondary meaning of the verb מָלַח, where integrity of the disciple of Jesus is lost and the result is disintegration (or moral decay), which is "not even fit for the manure pile" (Luke 14:34-35).