Is it professional to have a "cited by" section of a personal website?

For any well established research, they will usually have quite a large number of citations, which would make a "cited by" section would be exhausting to maintain and large beyond meaningful readability. A better route is to link to an database that actually engages in curating ones citations, such as Google Scholar.

I do, however, sometimes see people maintaining a collection of notable articles in non-scientific media. These are not normally picked up by citation databases and are valuable for showing that work is having impact outside of the scientific community, as well as (sometimes) providing a nice introduction to the work.

To expand on @jakebeal's answer: If you can still list all citations you have ever received, then the number of citations you have received is so small that the attempt reflects negatively. Most reasonably well established scientists have thousands of citations to their credit. It becomes impossible to list them all. The list also often grows by a dozen a week, more than you probably want to keep up.

As already noted, the solution lies in linking to a referencing service such as google scholar or MathSciNet.

Let turn this round....

People reading your website may be doing so because they are interested in your research topics. Therefore assuming you list your papers by topic, it could be very helpful for you to also list papers that expand on your papers in interesting ways, including explaining how they expand on your work in a few sentences.