If your work is art - bought for aesthetic pleasure, like paintings, sculptures, or books - you may dedicate it. Art is all about the artists' expression, their artistic personality; dedicating the work to a person they love is simply another part of that intimacy that an art viewer develops with the artists.
If you're writing a tool or a service, personal whims have little space, and more important than those are a more thoughtful design and a more professional feel.
This doesn't mean, however, that a web application cannot be art. Video games, for instance, are often considered art, web-based videos games included. So long as whoever sponsors the project agrees to it (assuming the role of an art patron - most likely you'll do it on your free time), the project can be made to feel like art, and in those cases there's place for personal touches such as dedication.