What is a word that describes someone who hurts people without meaning to?
Solution 1:
"Reckless", "careless" or "negligent" all mean roughly the same thing:
"Not giving sufficient attention or thought, especially concerning the avoidance of harm or mistakes." (Wiktionary)
The reckless person might hurt themselves or other people - that's ambiguous.
Solution 2:
You could you "clumsy" for hurting physically, and "tactless" for hurting emotionally.
See the following from subscription-only LDOCE:
tactless (adj.): someone who is tactless says or does things that offend other people or hurt their feelings, without intending to and without realizing that they have done it:
He made all sorts of tactless remarks about her appearance.
-> it's tactless (of somebody) to do something:
It was a bit tactless of you to start talking about her ex-boyfriend.
clumsy (adj.):
moving or doing things in a careless way, especially so that you drop things, knock into things, etc.:
A clumsy waiter spilled wine all over her new skirt.
a clumsy object is not easy to use and is often large and heavy
a clumsy action or statement is said or done carelessly or badly, and likely to upset someone:
David made a clumsy attempt to comfort us.
P.S. "insensitive" is another related word, however it doesn't exactly imply that the person is actually doing something hurtful, but just not noticing how the others are feeling hurt.