Could a candle theoretically melt iron?

No. The candle can only transfer heat into the iron as long as the candle is hotter than the iron. The temperature of a flame depends heavily on the chemicals being burned. A typical candle burns at about $1400\ ^{\circ}\mathrm{C}$. Iron melts at $1538\ ^{\circ}\mathrm{C}$. So, the iron would stop heating up at $1400\ ^{\circ}\mathrm{C}$ and not melt.

However, you're not that far off! Probably you could find a hotter than normal candle (mix sawdust into the wax, maybe?) and use it to melt a tiny piece of iron, like the tip of an iron needle. You might find an iron alloy that melts at a lower temperature, too.

Keep in mind that the equilibrium temperature will be lower than the candle temperature since the iron loses heat to the environment through processes such as radiation, as mentioned in other answers (see @Lelouch's answer). These loses can be reduced by doing things like enclosing your system in a shiny insulated box to reflect radiated heat back.

I'll try a simple explanation. Assume that there are no phase transitions initially. As you heat a body, its temperature rises, and it radiates energy into the surrounding space according to $$P = A\varepsilon \sigma T^4$$ ($\sigma$ is the Stefan Boltzman constant, $A$ is the surface area, $T$ is the temperature, $\varepsilon$ is the emmissivity).

Obviously, $P$ increases as $T$ increases. So, if before the bar reaches its melting point, $P$ becomes equal to the power input (from the flame), there will be no net flow of energy across the surrounding-bar interface (for if there were, $P$ would be greater than the input power, and more energy will be lost than gained, taking it back to the stable equilibrium point). At this point, notice that the temperature is constant, because any energy supplied by the candle is equivalently emmitted by the body (sort of dynamic equilibrium). Of course, this is only possible if the bar doesn't melt before this temperature is reached.

Is heat cumulative?

This is a bit of an ambiguous question, but yes it is "cumulative". If you put some heat into something, it will stay hot, unless that heat is taken out. Putting 10 units of heat at once is the same as putting 1 unit 10 times, again assuming that you can keep the heat insulated in the object.

The analogy here is a bucket with holes. Heat is like water you put in, temperature is the level of the water. It doesn't matter if you put the water in all at once, or in small bits, but as soon as you put in some water, it starts leaking out. So if you don't hurry up, you might never get enough water in, and the more water is in the bucket, the faster it leaks out.

Can I melt iron with a candle?

Probably not. I think the biggest problem is that heat flows from hot things to cold things. If the iron reaches the same temperature as the gas coming from the flame, it won't receive any further heat. If your target temperature (the melting point) is actually below this, you'll get stuck.

As it happens, the melting point of iron is about 1600C. The flame of a wax candle tops out at 1500C, but that is a tiny area in the center - the average temperature is 1000C.

You could solve this issue by using an exotic candle made of unusual materials. For instance, thermite burns hot enough to melt steel, acetylene torches can get you to 3500C. But when you say candle I assume you meant wax candle, and not a thermite "candle".

Even with a hot-burning candle, you still have the problem of small heat flux. You say:

the heat-losing rate is below the heat-getting rate

Sure, but that's easier said than done. If you had a magic iron that never loses heat, okay, but in real life the hot iron will heat up the air in the room (candle won't burn in vacuum), the hot air will float away, and in effect you will end up heating up the whole room because it's all in thermal contact. That's asking for a lot from a candle, but if you somehow had a candle with infinite fuel and had it burning indefinitely, then eventually you could heat the whole room above the melting point of iron (it has to be above because melting itself requires some heat input), and yes, it would melt - but we are now talking about a very extraordinary candle. Not to mention the mass of such a candle could act as a heatsink, or it could ignite in your firestorm of a room, so there is a whole host of practical problems.

So in conclusion, what kind of candle would it take to melt iron? You need:

• A fire hotter than 1500C.
• An insulated environment which can get to this temperature and stay there.
• A big enough candle to ensure that the rate of heat loss is always less than gain (remember that heat loss is faster when the temperature difference is larger).

This is kind of device is called a forge.