Sometimes the best way to overcome bias is through an emotional appeal. Below I interweave discussion of how emotional appeals can be used to overcome the bias corresponding to the identifiable victim effect and maladaptive resource hoarding instinct.

The beginning of the abstract to Paul Slovic's article "If I look at the mass I will never act”: Psychic numbing and genocide reads

Most people are caring and will exert great effort to rescue individual victims whose needy plight comes to their
attention. These same good people, however, often become numbly indifferent to the plight of individuals who are “one of many” in a much greater problem.

Eliezer has discussed this and related topics extensively, for example in Scope Insensitivity. See also the references listed at wikipedia under identifiable victim effect. How can we go about overcoming this bias? One answer is "by keeping it in mind and by teaching people about it." But while some people have the intellectual interest and ability to learn about the identifiable victim effect, others don't. Moreover, it's not clear that being aware of this bias is by itself very useful in overcoming it.

But reason is not the only means of overcoming bias.

Before I proceed I should make some disclaimers:

Disclaimer #1: Below I discuss overcoming biases that people have against donating money for the purpose of improving health in the developing world. In doing so I am not advocating in favor of developing world aid over other forms of charity. The case of developing world aid simply provides a good example. Most people who decline to donate to charities that improve health in the developing world do not decline to do so because they think they have a better cause to donate to.

Disclaimer #2: This posting is intended to advocate the use of emotional appeals specifically for the purpose of overcoming bias. Obviously emotional appeals can and are frequently used to create bias - this is not what I'm advocating. I believe that the video which I discuss below does more to overcome bias than it does to create bias for the typical viewer

Disclaimer #3: The use of emotional appeals strikes some as patronizing or manipulative. I do not view them in this light when they are used for the sake of aligning people's behavior with their values for the sake of good causes. I subject myself to stimuli with emotional appeals in order to keep myself motivated to do what I think is right. The video below is one such example. Another such example is the paragraph of Eliezer's One Life Against The World which reads

I agree that one human life is of unimaginably high value. I also hold that two human lives are twice as unimaginably valuable. Or to put it another way: Whoever saves one life, if it is as if they had saved the whole world; whoever saves ten lives, it is as if they had saved ten worlds. Whoever actually saves the whole world - not to be confused with pretend rhetorical saving the world - it is as if they had saved an intergalactic civilization.

Disclaimer #4: In referencing the topic of charitable donation, I'm not advocating that people encourage others to donate money to any cause to the point of becoming unhappy about it. A few months ago I wrote an article explaining my position on this matter here. I was interested to be reminded that Eliezer has made similar remarks in his video response to a question by komponisto.

The video titled The Life You Can Save in 3 Minutes does a great job of overcoming the absence of an identifiable victim effect without providing an identifiable victim and without explicitly mentioning the identifiable victim effect at all. The first 1:15 minutes make a case for the viewer donating money to save lives in the developing world. The text of the next segment of the video reads

(cue the skepticism)

Does aid even work?

Doesn't it breed dependence?

Isn't it wasted?

And poverty is bottomless!

Eternal!

Infinite!

Isn't it all pointless?

In doing so the authors of the video are channeling a common reaction to appeals for developing world aid. It's rational for viewers to have the concerns mentioned above. As the video comments next:

There are long answers to these tough questions.

Developing world aid is a very tricky business and a high level of vigilance is required to make sure that it goes well. But at the moment it appears to me that there's a bottom line that it's possible to greatly improve people's lives by donating money to certain charities working to save lives in the developing world. See GiveWell's International page for more information.

More to the point of this post, in practice there are problems of people:

Raising legitimate doubts as to the value of developing world aid and then failing to follow up on determining whether these doubts are well grounded.
Asymmetrically focusing on potential negative unintended consequences of developing world aid rather than potential positive unintended consequences of developing world aid.
Imagining that donating money would correspond to a greater sacrifice of material resources than it actually does.

I view these observed behaviors as being in line with Yvain's Conflicts Between Mental Subagents: Expanding Wei Dai's Master-Slave Model: it seems as though these lapses in reasoning arise from modules of our brain which were designed to allow us to preserve our self-image as a good person while declining to relinquish material resources. I agree with Yvain's Would Your Real Preferences Please Stand Up? posting that there's no meaningful sense in which such lapses in reasoning should be viewed as revealing our preferences.

As I say in Missed opportunities for doing well by doing good I suspect that:

Our tendency to hoard material resources is largely maladaptive and doesn't improve our lives very much at all.
The psychological benefits of donating money to help others are considerable and systematically undervalued

So it looks highly desirable to help people overcome their irrational hoarding of material resources, overcome the psychic numbing attached to the absence of an identifiable victim, and see the relative costs and benefits of donating in clear terms. The creators of The Life You Can Save in 3 Minutes do a great job of this in the subsequent portion of the video. After the text "there are long answers to these tough questions" the video shifts into a different mood and the text reads:

But here's the short one:

What if your daughter was the "drop in the bucket"?

Real lives are saved every single day. People with real names whose families weep with joy to see them still alive.

If you were one of those people you wouldn't think it was pointless.

This serves to dispel the absence of an identifiable victim and make the true benefits of donating (or equivalently, the true opportunity cost of not giving) psychologically salient. [Edit: At least sometimes, but see byrnema's comment and my response.] A portion of the remainder of the video makes low cost of giving to the donor psychologically salient. Key to the video's effectiveness is its use of music and visual imagery. The video is well worth viewing in entirety.

As I said above, sometimes the best way to overcome bias is through an emotional appeal. This is especially so when communicating with neurotypical people whose minds are naturally drawn toward emotional detail rather than logical detail.

Members of the Less Wrong community who are interested in this topic might find it useful to study the research of Deborah Small.