My daughter's been in dance for 7 years now. She went through a phase at about this age about not wanting to go as well. Ended up that there was a girl in class that was constantly picking on her and she just didn't like being there because of it.
Talk to the teacher and see how cooperative she is in class. Kids have a habit of being more dramatic with their parents around. If your "pick up friend" says she's OK after class, then it could be as simple as separation anxiety.
What's her day like? I also notice (even now at 11 yrs old) that when mine is tired or stressed because of something at school (big test, general bad day, etc) she can be a bit dramatic. Mine also used to figure skate in the mornings before school and it was getting difficult for her to be so overly active. We cut out the early mornings and added two more dance classes. Seems she's much more stabile now (for lack of a better term!)
If her feet are hurting from tap, be sure to check out her shoes. Maybe another size would be better, or maybe she needs socks to keep from rubbing/pinching, or something as simple as a planter's wart, or a split toenail could be the culprit.
Don't give up yet. Work with her. Put her in the position of control- or atleast make her think she's in control. Do you want to wear the pink leotard or the blue one? Should we eat a snack at the studio or wait until we get home? Do you want to take your <favorite stuffed animal> to the studio to watch you tonight? Try to encourage her, yet guide her into trying again.
And, for what it's worth - I've always required my kids to follow through with their decisions. But, I'm the meanest mom on the block too, and each kid is different! Especially something with a long schedule and larger committment as dance. But, I started them out with things that had much shorter durations like an 8 week swim class or a 10 week karate class. Wasn't quite as painful or stressful for us, and they understood that a committment was a committment to the end.
Good luck - I know it's not always easy! But, being the mom is be best job in the world!