At initial stages, no, it is not important until your site and business cards design looks nice. So for business cards, as a startup, don't worry if your logo at initial stages is just a Arial Black or Helvetica Bold font name of your project. Nobody will mind or notice.
Logo or not, you want to look professional for the best impression.
Does your website look professional as it is without a logo? If so, fine. If not, add a logo to it.
Whatever you have on your website, you should reflect it in your business card. If you decide your website is fine without a logo, then your business card should be fine without a logo as well.
When you change your website (to add a logo or change style), change your business card at the same time to reflect it.
Logo's aren't useful if not implemented correctly. And definitely do not use a generic logo at all.
However, a quality logo that depicts your business can help. So while its not necessary for your first batch, definitely consider getting one soon.