The Church's Stance on Debt
The official stance of the LDS Church regarding debt is summarized in the following statement from the Church's website:
Since the early days of the Church, the Lord's prophets have repeatedly warned against the bondage of debt.... When it is necessary to incur debt, such as a reasonable amount to purchase a modest home or to complete one's education, the debt should be repaid as quickly as possible.
There are no caveats or exceptions for Church officers or other leadership.
Concerning the Verses in Doctrine and Covenants
I can't find any specific, authoritative statements regarding the meaning of the verses referenced from Section 64. Here is how I read it as a member of the LDS Church.
First we have to back up a little bit and read the preceding verse:
26 And it is not meet that my servants, Newel K. Whitney and Sidney Gilbert, should sell their store and their possessions here; for this is not wisdom until the residue of the church, which remaineth in this place, shall go up unto the land of Zion.
The Lord is speaking to two specific members of the Church in these verses: Newel K Whitney and Sidney Gilbert, both of whom ran some small businesses on the behalf of the Church. The three verses quoted in the question are actually three separate but related statements directed at the two businessmen:
27 Behold, it is said in my laws, or forbidden, to get in debt to thine enemies;
28 But behold, it is not said at any time that the Lord should not take when he please, and pay as seemeth him good.
29 Wherefore, as ye are agents, ye are on the Lord’s errand; and whatever ye do according to the will of the Lord is the Lord’s business.
Verse 27 is a clear denunciation of debt, which is still taught in the Church today. This may have been an important reminder for the two businessmen.
Verse 28 is, I believe, an almost parenthetical reminder to the two men of their own position before the Lord; a rephrasing of the Lord's proclamation, "My ways [are] higher than your ways, and my thoughts than your thoughts" (Isaiah 55:8). After all, He has given men everything, and men are eternally indebted to Him; therefore, anything man does for Him does not require recompense (though He may give it anyway in His love and mercy). He is effectually saying, "Remember who is in charge here."
Verse 29 is similar to verse 28 in that it reminds the businessmen that they are working under the direction of the Lord. It also emphasizes that they are His agents doing His business, as long as they follow His will.
Each verse conveys a distinct idea. The Lord instructs these two men to: 1) avoid debt; 2) remember that the Lord is God; and 3) remember that you are an agent of the Lord, and that you should follow His will.
I do not read in these verses that the Lord is giving permission to his officers to refuse paying their debts. To my knowledge, Church officers have never refused to pay their debts by referencing these verses, and I don't see any reason to believe that the Church interprets these verses in that way.