Wouldn't backwards time travel break the law of conservation of mass? We know from the law of conservation of mass that the amount of mass in the universe is constant. Suppose there were a way for a person to travel backwards in time. Let's call this mass $m_t$ for some time $t$. Assume you are traveling backwards in time from $t_n$ to $t_0$ where $n$ is the amount of time (let's say the unit is seconds) since the time traveled back to. This allows for negative subscripts of $t$. If the law of conservation of mass is true, then it is true for all times. Then $m_{t=-1}=m_{t=0}$. However, since you have traveled through time to $t_0$, you have added your mass to $m_{t=0}$. That means that according to conservation of mass, a quantity equal to your mass has been subtracted from something else. But mass doesn't spontaneously disappear in order for time travel to occur, so that's impossible. Doesn't it follow that by the law of conservation of mass, backwards time travel (and by similar logic but in reverse, forwards time travel) is impossible?
 A: I can't give a good answer, but I can give a layman's bad one.
Time travel is very complicated and probably impossible.  I've read a few dumbed down articles about it and it's one of those things that just doesn't seem to work, even in theory, using what we know of quantum mechanical and relativistic models.  There's no shortage of articles on this, search for "is time travel possible" threads on this board and elsewhere on the internets.
When you say this:

We know from the Law of Conservation of Mass that the amount of mass
  in the universe is constant.

More details covered in this question, but in short, that's not strictly true.  The Universe might not have constant mass.   But if we ignore all that, there's still ways around the problem.
One is, mass-energy equivalence.   If you send something back in time, theoretically that would take some energy.   Where would that energy come from?    To make it work, you might need to borrow energy from the past, so that the mass-energy in the past stays the same.    How can you borrow energy from the past?  Strangely, that might be possible, at least on a quantum level, sorta.   Besides, if you send an object into the past, borrowing energy from the past to maintain balance might be required for balance. 
While pure science fiction, the way they handled cross dimensional transport in the TV-show fringe is that they had to swap equal amounts of mass.   If you send a person back in time, you might need to take back the mass equivalent (perhaps quantum spin and charge equivalent) back in return, so both times retain not only the same mass and energy but the same charge, spin, number of protons, and neutrons - all is preserved.
Such ideas aren't really physics since time travel of this sort probably isn't possible.
A: Mass is not conserved. That is the first thing. When you look at relativity, and space-time, you have to consider the posibility that our universe is not ne in which time passes and concepts of past, future or present are absolute. They are not. Your future might be my past. You have to consider the posibility that all space-time exists not just space in this now moment. So should you travel in some way to some other space-time point where there is another you, you would just transfer your mass to that space-time point. If all the moments and all the positions exist it does not matter where you are exactly.
Of course, this is just waving hands and chating..but it came across.
A: True, in General Relativity, the concept of conservation of energy becomes very muddied.  But that is irrelevant.  In Goedel's rotating universe, for example time travel even though always forwards, gets you into the past eventually.  But since you were already there, there is no problem with the masses.  
In special relativity, with a positron considered as an electron travelling backwards in time, conservation of energy is preserved because when the electron, initially travelling forwards in time, decides to scatter backwards and become a positron, what makes it do this? A very energetic photon.  So the energy is conserved since the vanishing electron--positron pair are compensated for by the energetic photon.  Mass-energy is conserved.  So it is not too hard to get an electron to travel backwards in time.  But the more massive the particle, the more energy is required to get it to scatter backwards.
Since special relativity is approximately true, these considerations are approximately true in general relativity as well:  conservation of mass makes time travel more expensive.
A: 
We know from the Law of Conservation of Mass that the amount of mass in the universe is constant.

False. There is no law of conservation of mass. Mass seems to be conserved in our everyday units because the laws of special relativity , E=m*c^2, start to apply at large kinetic energies which are not accessible in everyday life.  Experiments with accelerators and cosmological observations show that what is conserved in regions of space with low gravitational forces is the total energy of the system expressed in special relativity terms. When one gets to large gravitational masses General Relativity laws hold, and there  the concept of energy has to be redefined, and its conservation is not a law for cosmological scales. 

Suppose there were a way for a person to travel backwards in time.

There exists no way within our standard physics that this can happen, as special relativity does not allow crossing between directions of time, the light cone is defining.

Let's call this mass mt for some time t. Assume you are traveling backwards in time from tn to t0 where n is the amount of time (let's say the unit is seconds) since the time traveled back to. This allows for negative subscripts of t. If the Law of Conservation of Mass is true, then it is true for all times.

There is no law of conservation of mass. As you will realize if you study a bit this is a science fiction scenario, not accessible to physics calculations.

Then mt=−1=mt=0. However, since you have traveled through time to t0, you have added your mass to mt=0. That means that according to LoCoM, a quantity equal to your mass has been subtracted from something else. But mass doesn't spontaneously disappear in order for time travel to occur, so that's impossible. Doesn't it follow that by LoCoM, backwards time travel (and by similar logic but in reverse, forwards time travel) is impossible?

I cannot follow your logic of making mass a function of time, but it is not important as backward in time moving masses are unphysical, and forward in time moving masses happen all the time. In our low energy everyday framework masses are conserved and do not change with time.
The rest of the answers are on the science fiction scenarios, which are fun, but have little to do with calculable physics, except some temrinology.
A: There are some incorrect terms in this question, but the idea you bring up is an interesting topic. Despite that we cannot travel in time, this idea still proves why we can't.
Also, as many have pointed out, there is no such thing as conservation of mass, but there is such thing as conservation of energy (in a system).
In the following case our system is the universe:
Et = Et'
This is the first law of thermodynamics. Following this law, Einstein's Theories of General and Special Relativity in conjunction with the Lorentz transformation (based on the Lorentz-coordinate system) takes us to an idea of conservation of mass.
E = mc2/√(1-v2/c2)
From this we derive:
mtc2/√(1-v2/c2) = mt'c2/√(1-v2/c2)
All of this puts together an ideal of total conservation of energy. When this all loops together, the transfer of energy from one point to another is illegal, but I can think of one exception.
Suppose your time traveling box (achem TARDIS) goes back 100 years. In this trip, it takes the energy of it at present time back with it to 1915 (the year Einstein supposes the theories of relativity). The loophole I found would occur because the energy would travel forward in time and eventually get caught up the present.
The biggest problem I find is quantum bootstrap paradoxes (explanation of the bootstrap paradox here). Let's say a quantum particle's theoretical spin state is revealed (in coordination with one on the time traveling box (you need some insight in quantum mechanics for this)) they are now connected. Now we go forward and run in to a similar problem as the bootstrap paradox, just quantum.
All in all, there are many more issues (e.g., observational light cones) with time travel, but that is a good and legitimate question I have asked myself.
Works Cited
Einstein, Albert, Hanoch Gutfreund, and Jürgen Renn. Relativity: The Special & the General Theory: 100th Anniversary Edition. Princeton, NJ: Princeton UP, 2015. Print.
A: In all technicality, the mass that makes up you came from other sources that would already exist in this time period, but you could never visit a time you existed. 
Say you went back to your parents' wedding. The egg and sperm from your parents exist, as well as all of the nutrients that would grow into you. This means it would not break the rules of the Conservation of Mass. It would also be plausible that this means your body would be formed from this mass, meaning if you stayed past the point where you were born, the mass needed to make you wouldn't be there and you would never be born. BOOM! PARADOX!
Say when you go back in time, this paradox is avoided by only sending your consciousness and not a physical form. That would solve all possible paradoxes by making it impossible to interact with physical objects, while also explaining supernatural activity. Unfortunately, this scenario means that your body is still aging at the average speed. BUT WAIT!
If the time travel was only inflicted on your mind, it would progress as if it was a dream. The average REM cycle is 15 seconds. The average dreams take up approximately one days time in dream land. This means that you would live four days during every minute. So technically if this form of time travel was possible you would live the entire month of February in the past but only age seven minutes.  
