I could swear this wasn't a problem before, but text seems to have simply stopped appearing in my rendered SVGs when it's set to flow into an object.

For example, if I do the following...
1) Create a new Inkscape doc
2) Create a rectangle
3) Create some text
4) Select rectangle and text
5) Text->Flow Into Frame

...the result is the expected text inside the rectangle, when viewed in Inkscape. But in both Chrome and Firefox the rectangle appears empty (though inspecting the source shows that the text is indeed there).

Does anyone have any idea what might be causing this? I've attached an example.

The flowed text is not part of the svg 1.1.
Until svg 1.2 will be ready, browsers wont render flowed texts right.
You should convert the flowed text to normal text if you want it to appear in chrome.
And, to avoiding any problems from "missing fonts", you better convert the text to path.

Thanks for the quick reply. So if I'm creating, say, a flow chart, what's the best practice for having a box or other shape with text "attached" to it? Is there any browser-compatible way to avoid having to manually "shape" the text so it doesn't overflow the box?

Don't know what exactly you want to achieve with the linking.
If you create an svg with the text, you want it to be able to change interactively through a browser?
Why simply positioning wouldn't make it?
There is an option to adjust the position precisely through the align and distribute panel, Ctrl+Shift+A.

It is surely possible to make it interactive in a webpage, but may require some coding skills.