No! It's very common to be told that dance shoes should be extremely tight, but in my experience, that's wrong. There's no way you can dance well on painful or numb feet and you can damage your feet that way (and maybe end up not being able to dance at all!)

Dance shoes should be snug but not tight. The point is that your foot should not slide at all within the shoe in any direction. That's all that's needed.

If you're a woman, depending on the shape of your forefoot and toes, you may have real problems with how pointed most dance shoes are. Often, they really squeeze your toes together. If they are so tight that they are bending your big toe in toward your other toes, you are just asking for foot problems, I think. In fact, I found that I needed to go UP a size from my street shoe size to get a toe box that would allow my toes to lie straight in the shoe. I use a heel grip to keep my narrow heels from sliding and I dance without pain or damage to my feet.

Dance shoes should be snug but not too tight. Dancers should try different shoes to find the right shoes. Even the same size shoes come in differnet shape, width, and support. I often carry several pairs of shoes of different sizes, because my feet change in size, dependent on the time and the day. I also carry different socks and shoe pads, so that I can adjust the fit of my shoes accordingly. Also, misshaped or worn shoes can cause serious damages to your feet. Dancers must take care of their feet.