As far as I understand Martin Luther according to the Catholic and the Eastern Orthodox Churches possessed the Apostolic succession while he was in the Roman Catholic Church (he was a priest, after all). However, now both Luther and all his followers are considered by both Catholic and Eastern Orthodox Church as those who don't posses the Apostolic succession. So, according to them, at what particular point Martin Luther lost his Apostolic succession?
When Luther was excommunicated for insubordination on January 3, 1521, he lost the ability to say he was in the chain. You cannot succeed someone who has disowned you.
Note the date to which most people would ascribe the "beginning of the Protestant Reformation" (a bad term since it was really a process that had been going on since the 1300s!) is October 31, 1517, when Luther nailed his 95 Theses to the door at Wittenburg Cathedal—a document outlining his beefs with the Catholic church, as it was acting at the time. The heresies for which Luther was excommunicated are not generally given a specific name (unless if you want to call Lutheranism or even Protestantism a "heresy").
According to the Catholic Church, the short answer is that he never lost it. In fact, in the strict sense, Martin Luther never participated in apostolic succession to begin with.
Apostolic succession refers to the fact that all bishops can trace their holy orders all the way back to the Apostles. That is, the Apostles ordained certain men bishops (or what we would call bishops today) by the laying on of hands, and these, in turn, ordained other bishops, and so forth. (See Catechism of the Catholic Church [CCC] 77-79, as well as 1 Timothy 5:22, which is an example in the Scripture of where this kind of succession occurred.) Thus, at least in the strict sense, apostolic succession does not apply to presbyters (i.e., priests).
Since Martin Luther was only a presbyter, not a bishop, he did not (at least in the strict sense) participate an apostolic succession. For example, he never had the capacity to ordain another man bishop. (Only a bishop can confer the Sacrament of Holy Orders. Hence, Luther could not ordain deacons or priests either; see CCC 1576.)
What the O.P. may have been thinking of was the status of Martin Luther’s holy orders.
The Catholic Church teaches that the Sacrament of Holy Orders confers a permanent “mark,” called a character, on the soul of the person who is ordained. (See CCC 1582; it is a “mark” in a metaphorical sense, of course: the character consists in the capacity of the ordained minister to perform the ministry that he is called to, especially the capacity to confer certain sacraments.) This character is permanent, and no power on earth can remove it.
As a consequence, Martin Luther remained a validly ordained priest to the end of his life. His status in the Church became irregular when he was excommunicated in 1521, but that in no way affected his holy orders.
The reason that Lutherans (as opposed to Luther himself) do not maintain Apostolic succession is that they did not attempt to continue it. Luther himself, as I mentioned, was only a presbyter, so he could not ordain new bishops, priests, or deacons.
Doctrinally, Luther largely disavowed the concept of the ministerial priesthood, so he did not seek to continue it. In any event, in order to do so, he would have had to have the cooperation of a Catholic bishop (or at any rate, a validly ordained bishop, such as an Orthodox bishop), which was not likely during his lifetime. Consequently, although many first-generation Lutherans were priests, their holy orders died with them.
1521 might be the Roman Catholic answer to this question, but the Eastern Orthodox answer is a bit harder to give. The Roman Catholic Church lost its apostolic succession in splitting off from the Orthodox Church (remember, this is the Orthodox perspective).
As late as 1274 and then at 1438, there were still attempts at reunion between the Roman Catholics and the Orthodox. But the reaction of the Orthodox people to the attempted reunion in 1438 made it clear that by that time the schism was a fait accompli.
That being the case, in the Orthodox view Luther was already in a church that had lost its apostolic succession, and his actions did nothing to remedy that.
Interestingly, some of the Reformers did have contact with the Patriarch of Constantinople, and explored the possibility of reconciliation between the two groups. Here's an article that gives a good summary of what took place.