How do non-LDS Christians respond to the LDS argument that Joseph Smith died as a Martyr for God, just like the Apostles?

Here are several thoughts that may occur to a non-LDS Christian when they encounter this argument.

1 - The fact that someone dies for their beliefs does not make those beliefs true. That the Apostles died for their faith does not prove Christianity to be true, but it does suggestion a question, "Why did these 1st century Jews give their lives for a dead Messiah who never sought to overthrow Rome?" If we consider the following historical realities, it's not hard to see that the martyrdom of the apostles at least should cause us to pause and wonder at this massive paradigm shift. And the resurrection is a powerful argument for that paradigm shift.

a) Jesus had already died - many Jewish Messianic movements in the first century simply evaporated when the supposed Messiah died - why were these Christians persisting even to the point of dying for their belief in a resurrected Messiah if their Messiah was dead?

b) 1st century Jews believed in a Messiah who would overthrow Rome and establish the Messianic Kingdom. Jesus did not set up any form of earthly Kingdom. Moreover, Jesus died a shameful death on the cross. In addition, the Jews did not believe in a resurrection before the end of the age. For numerous devout Jews to give up their lives believing in a Messiah who suffered a shameful death, rose from the grave before the end of the age, and set up a spiritual rather than physical Kingdom, there must have been a powerful event that caused such a paradigm shift.

In contrast to the Apostles, it's not as difficult to understand what Joseph Smith might have died for - he had built a small earthly kingdom where he married many women and wielded significant political influence. There is not the same kind of massive counter-intuitive paradigm shift and so the argument from Smith's death to his witness is simply not parallel to the argument from the Apostles' martyrdom to the resurrection.

2 - Joseph Smith did not teach the same Gospel as Jesus and the Apostles, or the historic Church, so it is not possible that his death functioned the same way as that of a martyr who was faithful to the teachings of Christ.

3 - It is questionable whether Joseph Smith died as an innocent martyr or because he was oppressing others. Even if we take the version of events listed in the link you provided - Chapter 46: The Martyrdom: The Prophet Seals His Testimony with His Blood, paragraph 530 - we can see that the Mormon version of the story goes to great lengths to make Smith sound innocent (see bold text) when they clearly know the circumstances make him appear guilty. It is easy to see how Smith might have been abusing his power to silence his critics, who were upset about his promotion of polygamy and his use of power, among other things. As 1 Peter 4 notes, if Smith suffered because he himself was misusing power or a wrongdoer in some other way, then his death does not qualify as martyrdom.

On June 10, 1844, Joseph Smith, who was the mayor of Nauvoo, and the Nauvoo city council ordered the destruction of the Nauvoo Expositor and the press on which it was printed. The Nauvoo Expositor was an anti-Mormon newspaper that slandered the Prophet and other Saints and called for the repeal of the Nauvoo Charter. City officials feared that this publication would lead to mob action. As a result of the action by the mayor and city council, Illinois authorities brought an unfounded charge of riot against the Prophet, his brother Hyrum, and other Nauvoo city officials. The governor of Illinois, Thomas Ford, ordered the men to stand trial in Carthage, Illinois, the county seat, and promised them protection. Joseph knew that if he went to Carthage, his life would be in great danger from the mobs who were threatening him. Chapter 46: The Martyrdom: The Prophet Seals His Testimony with His Blood, paragraph 530

1 Peter 4:12-16 (NIV)

Dear friends, do not be surprised at the fiery ordeal that has come on you to test you, as though something strange were happening to you. 13 But rejoice inasmuch as you participate in the sufferings of Christ, so that you may be overjoyed when his glory is revealed. 14 If you are insulted because of the name of Christ, you are blessed, for the Spirit of glory and of God rests on you. 15 If you suffer, it should not be as a murderer or thief or any other kind of criminal, or even as a meddler. 16 However, if you suffer as a Christian, do not be ashamed, but praise God that you bear that name.