1) It's a one-sided practice that maintains gender roles. He can and must do things for me but I can't reciprocate. I like to be able to do nice things for a guy - open doors for him, pay for dinner, drive, etc. - without him refusing or feeling emasculated.
2) There's usually an expectation that I will reciprocate in some other fashion. I keep hearing that dating is legal prostitution because the guy pays for things and he gets companionship and/or sex in return. Even if the guy doesn't see it that way, there's still an expectation that I will do "girlfriend" things, assuming the "woman's" role in the relationship, since he's assuming the "man's" role.
3) It gets in the way of equality. If I hear one more guy complain that women are hypocrites because they want equality but still expect chivalry I'm going to scream. I highly doubt there are that many women who want both - most women I know are either feminists who don't like chivalry, or anti-feminists who would rather maintain their status in society so men will keep doing things for them.
Which leads me to my 4th issue with chivalry:
4) Chivalry makes women feel okay with inequality. I'm 90% convinced that's part of why it was implemented in society, so women will feel superior, and so women will be okay with how things are. As long as men keep doing nice things for them and treating them like princesses and queens, they will be okay with gender inequality and won't question it, let alone fight it. It's similar to the feminine mystique in the 1950's, make the life of a housewife look charming and irresistible so women will love their status as second class citizens.
5) It's a simplistic approach to dating. We're stuck on this idea that a "good boyfriend" foots the bill and gets the door, rather than setting higher, more complex standards for what makes a good partner. Doing nice things for a girl mean very little if they're done out of obligation. Paying for dinner means nothing if he's condescending towards me during the meal and doesn't listen to what I have to say. Getting the door for me isn't nearly as important as how he treats me when we're inside.
6) It's makes a lot of assumptions. It assumes that every couple has both a cisgendered, heterosexual man and a cisgendered, heterosexual woman. It also assumes that the man has more money, is stronger, and more able-bodied. This doesn't do much for a guy's self-esteem if he can't afford to take a girl out on a date or buy her presents, and it really doesn't help a guy's self-worth if he's handicapped and can't open the door for his date, or if he's vision impaired and can't drive her to the restaurant. I definitely think it's a problem when guys feel emasculated just because they can't do "the man's job" in the relationship.
Now, I have no problem with a guy getting the door if he gets there right before me. It's when he sprints in front of me to get it, holds the door when I'm still fifty feet away, or doesn't let me hold the door for him that I get a little peeved. I have no problem with a guy paying for dinner, or lunch, or breakfast, or whatever, and I may let him pay for most meals if he's making more money than me. I'm not a fan of going Dutch most of the time, usually I'll only go for it if we're just friends. But if he insists on paying for ALL the things and never lets me reciprocate, there's a problem. And yes, I'm aware that some servers will judge a guy if he doesn't foot the bill, I think that's garbage.