In Matthew 16:19, we read as follows:
“And I will give unto thee [Peter] the keys of the kingdom of heaven: and whatsoever thou shalt bind on earth shall be bound in heaven: and whatsoever thou shalt loose on earth shall be loosed in heaven.”
This saying seems to be patterned on Isaiah 22:22 (I shall quote verses 20-23 for context):
 And it shall come to pass in that day, that I will call my servant Eliakim the son of Hilkiah:
 And I will clothe him with thy robe, and strengthen him with thy girdle, and I will commit thy government into his hand: and he shall be a father to the inhabitants of Jerusalem, and to the house of Judah.
 And the key of the house of David will I lay upon his shoulder; so he shall open, and none shall shut; and he shall shut, and none shall open.
 And I will fasten him as a nail in a sure place; and he shall be for a glorious throne to his father’s house.
Apart from what to Latter-day Saints will surely resonate as profound temple imagery in these verses, we can clearly see that verse 22 provides the pattern for the Matthean saying. In Isaiah, the imagery is of the vizier or master of a house, who possesses the key to the house and has absolute control over ingress and egress. Matthew also uses the imagery of the key to represent authority (putting the word in the plural; where a slave might have a key to the gate, only the lord of the house has all the keys) and also uses a similar type of antithetic parallel expression. But rather than sticking with the key metaphor, as Isaiah did, Matthew (or Jesus through Matthew) mixes his metaphors by using the Greek words dein [“to bind”] and luein [“to loose”]. Rather than referring to locking/closing and unlocking/opening doors with a key, these words are the Greek equivalents of Rabbinic technical terminology, the Aramaic ‘asar and sheri (Hebrew hittir). The words express the verdict of a teacher who declares some action as “bound” (prohibited) or “loosed” (permitted). There are numerous examples of this terminology in later Rabbinic literature, where, say, the school of Shammai will declare some action “bound,” but the school of Hillel will declare the same action “loosed.” This usage is also attested in the Targums, and even in Josephus.
As to what binding and loosing mean in this context, there are various theories. Probably the most common is that Peter and the other apostles were given plenary authority over doctrinal and disciplinary matters. Some, influenced in particular by the parallel in Matt. 18:18, see this as a power to ban members from and readmit them to the community. Others, influenced by the later parallel in John 20:23, see this as a power to forgive sins. What is clear is that the apostles were not acting in the technical rabbinic sense of that terminology; rather, binding and loosing are in some way metaphoric for power and authority the apostles were to exercise.
Of course, our own LDS reading of the passage equates “binding” with “sealing.” Compare Isaiah 8:16 (or 2 Nephi 18:16):
Bind up the testimony,
seal the law among my disciples.
where the terms “bind” and “seal” are used in parallel.
With that background, we may now consider the topic of this post. The expression “shall be bound in heaven” and its parallel “shall be loosed in heaven” are formed by a very rare construction in Greek, the periphrastic future perfect. That is, the future middle indicative verb estai “will be” works together with a perfect passive participle dedemenon “has been bound.” If we understand the future perfect as referring to time, the text literally says “whatever you shall bind on earth shall have been bound in heaven, and whatever you shall loose on earth shall have been loosed in heaven.” That is, the action in heaven, on this theory, would precede the action on earth. Some are troubled by the implication that earthly actions can determine heavenly actions, so they prefer to understand the future passive temporally, which does indeed avoid such an implication. Unfortunately, the temporal reading leads to the opposite implication, that earthly actions are predetermined by heavenly actions. Advocates of this view recognize the problem, and attempt to soften their interpretation as meaning more or less that the apostles should seek the will of heaven in taking their actions. While perhaps a laudable principle in and of itself, the temporal reading cannot be limited to this principle alone and eventually leads us to some sort of Calvinistic determinism.
The deterministic reading of this passage presumes that we must take the future perfect temporally, but such is not the case. The Latin Vulgate mirrors the Greek, but very few English translations give a temporal force to the future perfect (although some would argue that this is simply a case of pervasive mistranslation.) For instance, the NAS, which used to read “whatever you shall bind on earth shall have been bound in heaven,” now reads “whatever you shall bind on earth shall be bound in heaven.” Verbal aspect is a difficult subject about which we are continually learning. (For instance, when I was in school I was taught that an aorist tense is punctilear, referring to a specific point in time; that view has now been universally abandoned.)
Given the rarity of the construction, it is helpful to consult the standard classical grammar of Smyth, who discusses several different uses of the future perfect. Essentially, a future perfect denotes a future state resulting from a completed action. When stress is laid upon complete fulfillment, the future perfect may imply rapidity, immediate consequence, or certainty of action accomplished in the future (citing a Greek example that could be translated “and it shall be done instanter“). That is, the sense could be something like “whatever you shall bind on earth shall immediately be bound in heaven” or “shall most assuredly be bound in heaven.”
The other way to read this, which strikes me as the more likely, is to emphasize the stative aspect of the perfect: “whatever you shall bind on earth shall stand as bound in heaven.” This translation emphasizes the finality of decisions made with apostolic authority.