The bigger issue isn't necessarily if the font is serif or sans-serif. It's whether the font is readable on the web. Verdana, for example, was designed for digital reading whereas Arial was designed for print. I believe (but don't quote me) that Georgia was designed for digital and TNR for print. In situations where there's a lot of content, using a digital-designed font will very often be a stronger choice than a print-designed one. If only because of how they scale (and, as a side note, please please please make sure you're setting font sizes with ems as your unit of measurement! ;-))
Directly related to your question, though, it is fairly conventional to use a serif for headings and a sans for body content, but that's really only because the list of true web-safe fonts is very limited. If you want a visual distinction between headings and body content, and don't want to use comic sans ;-), then your choices are basically serif heading and sans-serif body. Not because it's right or wrong, but because that's basically the only choices you have that have sufficient visual difference in the web-safe-font world. And, as was mentioned, it's often stated that sans serifs are better for longer reading in a digital format -- again, if only because of how they scale. Ideas about that are also changing as digital-designed fonts get better and more prevalent.
But no, it's not a rule and especially not with the font replacement techniques currently in use (and the upcoming web fonts that are going to explode in the next few years). It's more important to make sure its readable and supports your goals.
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