The harm is not in homeopathy itself, but appears when homeopathy replaces science-based medicine. If you have a serious illness and you choose to rely on homeopathy instead of real medicine, you may as well do nothing. If you turn to homeopathy to relieve, say, a headache, then little harm is done because maybe the placebo effect is good enough to "cure" you. I still say little harm, because every improvement (seemingly) brought about by homeopathy might make you turn to it faster and, in the end, rely on it to cure you of something dangerous.

There is, of course, absolutely nothing wrong with homeopathy itself - it's just water with a bit of alcohol and sometimes unpleasant sounding body fluids mixed in. So, apart from the occasional inclusion of arsenic or somesuch poison in the mix, it may well work as a placebo in certain instances. It's not heroin.

The problem is homeopaths themselves, those that direct the treatment, almost entirely unregulated and occasionally utterly reckless and with seeming total ignorance of scientific and pharmacological understanding. So, on one hand if you have, for example, an itch on your leg, and your homeopath friend gives you some concoction to pour in your bath each night and upon doing this for three weeks the itch magically goes away, then you've probably not done yourself a great deal of harm. However, if your homeopath friend advises you to stop pharmacological treatment for your cancer or serious heart condition and try his pharmacology-free method, and you then proceed to die from your condition very rapidly and unneccesarily, then the homeopath will have accented quite acutely the stark problem homeopathic methods present to informed medical diagnosis and treatment.

Here's a nice quote about homeopathy I found from The American Journal of Medicine, by Michael Baum and Edzard Ernst:

"The true skeptic therefore takes pride in closed mindedness when presented with absurd assertions that contravene the laws of thermodynamics or deny progress in all branches of physics, chemistry, physiology, and medicine."

Homeopathy acts as a gateway to stronger forms of woo.

Well you know what my answer to this question is going to be.

I agree with the other answers that there is a serious problem when homeopathy is substituted for conventional treatment, resulting in harm that otherwise would have been prevented. But I disagree that homeopathy is otherwise acceptable.

Homeopathy costs money, wasted money, to the user and/or to society indirectly through higher premiums or taxes if covered by private or public health plans. This money is received by quacks, who then do their best to discredit scientific medicine and promote the legitimacy of unproven techniques, thereby encouraging any number of other pseudomedicines that may be more directly harmful, as well as causing harm to the perception of legitimate science and medicine and decreasing public understanding of legitimate science. It's not a stretch to consider that believers in such bunk then, if elected to positions of power, are able to influence legislation and budgeting in such a way as to further harm science and propagate or support alternative medicine.

Finally, homeopathic "remedies" are not prepared by doctors or pharmacists for the most part, and thus it's quite possible that harmful components may be introduced during the preparation process. I simply put it out there are a possibility, I have no studies to back that up.