What does "have a heart that never hardens, a temper that never tires, a touch that never hurts." - Charles Dickens mean?

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It means:  don't lose your humanity.

I'm sorry to say that the one person who thinks this passage is clear as written (and those who "thumb up" in agreement) doesn't seem to actually understand it. This is why it does become a Herculean task to explain.

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It means never lose touch of the fact that others hurt the same as you, never lose your ability to fight for what is right, and never use your hands to hurt another.

Oh, for fcuk's sake, @AmulyaP, it's not "Herculean" at all. What Dickens wrote here is quite simple and easy to understand -- or it was before you obfuscated the hell out of it.

Explaining writings of Dickens is after all a Herculean task, however I'm making a small attempt.  

      The heart of everyone shouldn't be allowed to be stone-hearted , cruel which is incapable of understanding others' emotions , sentiments.A person with a soft/ kind heart can change any sad milieu into a tolerable one.

        Temper should be spontaneous like a running rivulet.

        Our touch should have a healing power when we offer solace to a grieved person.

Thanks.