Why does a battery die more quickly when more resistors are added to the circuit? I will be explaining what I think:
A battery acts like a pump which provides energy to do work on negative charges to move them towards the negative terminal, and hence creating an electric field. Now, due to this field, when a wire is connected to make a circuit, electrons move from negative to positive terminal, thereby making electric current. Now, if I put more and more resistors in the circuit, the electrons will have to do more and more work. Let me explain in more detail: Suppose there is only 1 resistor. Then, electrons will flow through it, they will loose some of their energy(to produce some desired effect like light or heat), but, because they are under the influence of the electric field, they will again gain some more energy. Now, if I add more resistors, the activity will be the same. Electrons will loose energy, and then again gain some through the electric field. Then, why does a battery drain faster in more resistors? Battery is just pumping charge from positive to negative terminal, what else is it doing? Why does it drain quicker? 
(For good explanation, viewers can consider resistors to be bulbs)
NOTE: Resistors are being added in series!
 A: If you add more resistors in series the effect will be the opposite of what you say: the battery will last longer. A battery has a certain rated capacity, written in mAh (milliamps times hours). Divide this capacity by the current you are drawing and you will get how much will that battery last in hours, at the same current draw. More resistance means less current, so you obtain more battery life. Think of this water analogy: how much will it take for a water reservoir to empty if you drain it? The more water you take out per time, the faster it empties.
A: The power consumed by your circuit determines how fast the battery drains. P = I * E: power (Watts) is found by multiplying the current (Amps) by the voltage (Volts). Since your battery has a (reasonably) constant voltage under normal operation, current is the variable here. 
I = E / R, amps = volts / ohms. If we combine these two equations, we get P = E ^ 2 / R. Since the voltage doesn't change, when the resistance goes up, the current and the power go down. So, adding series resistance to the circuit will make your battery last longer.
A: Case 1:
They are connected in series - the more you add the higher the resistance, the less current - longer life
Rtot = R1 + R2 + ... RN
Case 2:
They are connected in parallel - the more you add the lower the resistance, more current - shorter life
1/Rtot = 1/R1 + 1/R2 + ... 1/RN
Case 3:
Mixed series/parallel connection - who knows! Requires circuit analysis (such as Thevenin or Norton equivalent circuit)
