Is It A Good Idea To Get Into A Theological Debate With Mormon Missionaries?

There's no harm in a good debate as long as it doesn't degenerate into a slanging match and become insulting to any of the parties.

Even if you've absolutely no intention of agreeing with the Mormon faith it doesn't do any harm to learn, listen and exchange your views. Religious matters are very personal and individual and some people are incapable and or unwilling to listen to others. A respectful and open attitude is helpful if you are going to debate faith with others holding different belief systems.
For some people it's not a good idea because they can't tolerate others who hold different views. If you aren't religious it might be easier to debate with Mormon Missionaries because you might not believe anyway and therefore wont get offended.
It's perfectly possible for agnostics and atheists to debate religious matters and this category of people often study the philosophies of world religions.
It's probably best to set the guidelines before you agree to debate. If you explain that you are very willing to debate but do not want to be converted or have no intention of allowing it to be a conversion session then you should be on firm ground.

Some born-again Evangelical Christians find it hard to resist debating issues of theology and articles of faith with Mormon missionaries and may even let them into their home in the hope of such a debate. While it is certainly appropriate to question what Mormon missionaries are taught to tell potential converts during these meetings, there really is no reason to attack their faith, or to enter into a heated theological debate with them. You may very well find that these young men and women are reluctant to enter into a heated exchange with you and (as ironic as it may seem) they may not appreciate it if someone tries to proselytize them during these meetings.

The key is to remain respectful during these encounters and to voice your concerns regarding their church's teachings in a non-threatening way. If you find what they are telling you is unacceptable, then it is better to simply inform them that you have no interest in converting to their religion, instead of trying to discredit, or prove wrong the various articles of their faith. These missionaries are not really prepared to enter into deep theological debates, nor is that their purpose during such visits. If you try to enter onto the terrain of apologetics with them and if you question the core of their faith, very often they will simply tell you that they know that what they believe is correct, because they came to this conclusion through prayer.