Yes. The first potential downside to having a business blog is that someone has to do it - regularly. Each blog post is dated and time stamped.
This is a great from the reader's point of view; one of the appeals of blogs is that they deliver fresh content. However, if your blog doesn't have frequent posts, it becomes obvious really quickly. It's common for bloggers to post daily, or even make several posts during a single day.
You don't have to go this far, but a blog that hasn't been updated for a week is a bad user experience. From the blogger's point of view, a blog can be time consuming and yet another deadline pressure.
Second, a blog needs an engaging voice. It has to "sound" authentic. A business blog doesn't need a voice that's raw or ranting, as you'll find on some personal weblogs, but it does need to let the personality of the writer show through. If your blog reads like a business memo or corporate directive, people won't want to read it. For businesses that don't have writers on staff, or for business people who aren't writers themselves, this could be a downside.