This word is one of my least favorites. It has no place in intelligent conversation. There are four contexts in which I hear it often used: children, the elderly, non-human (typically animals), and romantically.
The non-human form of this word isn't too bad. Sometimes you see a puppy walking down the street and someone has to point it out to the group. It's a slippery slope, though. One day you tell your friend to look at the cute puppy, the next you're talking baby talk and using words like "cuddly-wuddly" to express your thoughts. Conversation takes a bad turn when it stoops to the level of "cute puppy admiration".
This form is a two sided concept. On one hand, you have the kid who has worked so hard to be like an adult and daydreams what being an adult is like. On the other, you have the teenagers or adults who have several times the child's life experience and who remember what it was like to be that age, but not quite as closely as they quite remember. It's quite entertaining to be on the older of the two sides, laughing at what cute things a child says or wears or even their looks like in general, but as a kid, it's infuriating. Sure, some kids can use their charm to get away with stuff, but imagine how degrading it is to call a kid "cute" when they want so desperately to be grown up. You're laughing at them almost constantly over things you should be explaining to them, and they have no idea why.
This particular form is insulting to the point of being revolting, although it is definitely the least popular. Have you ever seen a lady with more life experience than you do something that she believes is normal but you see as eccentric? Have you noticed an old man unaware of his age? These are people to be celebrated, honored, and respected deeply, yet I have occasionally heard them called "cute" in their endeavors. That's like the child laughing at the adult for doing taxes or driving a car. It's just not right.
The last form of this word is maybe more annoying than anything else, although it does seem to push objectification slightly. You can get a group of girls together and they'll talk about tons of wonderful things... but once you add a cute boy into the room, faces turn red, accusations are made, and those obnoxious giggles begin. Some girls cannot be dissuaded from talk of a topic that won't matter in the long run, and then other girls don't care about what a cute boy thinks about a girl who will be over his looks in a week. As soon as you start calling someone "cute", you also start down the path of objectification by dehumanizing them slightly in your head and making them an item to be acquired. Not good, people. Not good.
In general, calling any living organism "cute" seems to place yourself above them and oversimplify their existence. It doesn't feel good to be reduced to another person's method of amusement and it's a back-handed compliment- it uses flattery to insult someone further. (An example of such a compliment would be if I told you that you're really good at soccer... because for once in your life you can actually do something successfully. It seems like I'm being kind by complimenting your skill, but it hurts on the inside because I'm insulting the rest of your life where you've tried to make your own way.) Overall, using the word "cute" puts self above others and gives you a sense of superiority over those who often aren't sure of what's wrong with their actions.