The balloon filled with air should pop first. The reason for this is heat capacity: it takes a great deal of energy to heat up water, and a lot less to heat up air, so the rubber will get to the burning point and pop the balloon.

Here's why this matters: because water has such a high heat capacity, it can act as a colossal heat reservoir- or for that matter a colossal heat sink, to any system you may be considering. In nature, it goes beyond even that, however: the amount of energy required to vaporize water is almost six times again what is needed to heat it from freezing all of the way to boiling. This is why the Earth is a habitable planet: water evaporates from the oceans, carries it up into the upper atmosphere where it condenses out and falls back to the surface, but the water vapor condenses to form clouds and rain, it leaves the heat from the oceans up in the upper atmosphere where it radiates away into space. This effect controls the temperature of the Earth so that it is within the range in which life is possible: add more energy in,the water cycle just dumps it right back out. Also,water is a greenhouse gas (much more powerful than carbon dioxide), and this keeps the Earth from getting too cold.

In addition to its natural importance, there are a lot of engineering uses for water's high heat capacity as well. This is how an automobile radiator keeps the engine from melting itself: the water takes heat out of the engine, and then dumps it outside of the vehicle. The same principle is also used to keep nuclear meltdowns from occurring in a nuclear powerplant, too.

Whoever told you to "consider and think" was right. Simply put, it takes a lot more energy to heat water than air. This means that the oceans are effectively soaking up a lot of the energy associated with global warming and slowing the process considerably. This is debatable, but generally true. However, because water does not change temperature quickly or drastically, a few degrees of warming in oceans or lakes may have dire consequences to organisms who are adapted to a narrow range of temperatures that they experience throughout the year. One example of this is in the polar oceans, where the water temp doesn't change much (generally -2 to +2 degrees C). If the global ocean warms 1-2 degrees this is like it getting 20% hotter (or like Santa Barbara suddenly turning into Arizona and you not having an air conditioner). Water also cools much slower than air, so that in fall and winter lakes actually give off more heat than they take in. This also makes our climate "temperate"....look up the word as well as it's root: temper. Remember that the air is actually full of water (humidity) and so changes in air temperature can be less drastic in places where humidity is high. Ever been to the high mountains? It gets really cold at night and really hot during the day in part because of the low humidity of the air. Same is true in the desert. If you go camping, camp under a tree or next to a lake: trees breathe water in and out of the air so they have the same tempering effect as the lake on local temperatures. The balloon trick will be cool and hopefully will illustrate the point well

Thinking about natural water sources is a good way to think of the greater importance of this lab. If the water balloon pops last, why does that happen? There's a lot of heat coming from the flame, so where does it go? More importantly for the natural water sources, what happens to the heat once it gets into the water? Because of the fact that water can store a lot of heat energy, natural water sources are what we might call "heat sinks". If we have temperate weather, and then all of a sudden it gets really hot out, it might feel colder next to a lake because that lake can "suck up" a lot of the heat, leaving the air around it cooler. If we all of a sudden got a cold air front coming in winter, a lake that's been heating up for the whole summer could have enough stored energy to warm up the surrounding area by a few degrees (depending on the size of the lake, of course). This is also important in thinking about oceans - the reason why Britain is rather warm is because of ocean currents that bring warm Caribbean water up to Britain and warm it up. It's also the reason why coastal areas in southern California are a lot cooler than areas that are more inland - the water in the Pacific ends up coming from the arctic.

As far as the size of the balloon - more stuff in the balloon means there's more stuff to "suck up" the heat from the flame. Since water can hold a lot of heat energy, adding more water will make it take longer to heat the whole thing up. What about air? It can hold much less heat, so it probably won't make much of a difference how much air is in the balloon.

There are a couple of useful and interesting questions to consider here. One is the effect on the balloon material itself: which balloon will pop first, and why? A related experiment is to hold a plastic bag full of water over a flame. Most people are surprised to find that you can boil water in a plastic bag over an open flame, at least for a few minutes, as long as the flame or heat doesn't reach above the level of the water. That's a basic first experiment, which should lead you to ask: why does this work? How does water keep the bag or balloon from melting or popping? Does the material matter? Does the thickness matter? Does time matter? What is the temperature of the water, and does it matter? These are the kinds of questions that would allow you to create a hypothesis ("I think the thickness of the plastic matters because...") and test it with an experiment.

This question is important for a variety of reasons. For example, when your car engine burns gasoline, the temperature is about 2500 degrees C. But iron melts at 1370 C, yet your engine doesn't melt. Why? Also, nonstick cooking pans emit toxic gases when they're heated above 150-200C (roughly 300-400 F), but they're safe to cook most foods, especially foods with lots of water (eggs, vegetables, pancakes), even on a hot burner. Why would that be?

This leads to a kind of "big picture" question: what effect does the water play? This is where it becomes related to other things you mentioned, like water having a large heat capacity and affecting the temperature of things around it. If water soaks up a lot of heat, it's hard to heat or cool something else that's near the water. So objects in the water, or places near the water, tend to have the same temperature as the water, whether that's warmer or colder than normal.
Will it matter how big the balloon gets when you blow it up or fill it with water?
I wouldn't want to give away the answer. :-) The best thing to do is make a guess, write down what you think will happen and why, and try a few balloons: air only, water only but half filled, water only and extra filled. Always try to prove your idea wrong--that's the scientific method.

For safety's sake, try to do these experiments someplace where you won't spread a fire or get scalded if the bag breaks suddenly or catches fire. I find that a campfire or small fire in a BBQ works well, with a garden hose or big water bucket nearby in case anything goes wrong. Have fun!