I've done some more troubleshooting, and I've identified the cause as well as a solution. The solution allows Stack Exchange login to continue to work properly, without disrupting Adblock on other sites.
Diagnosis: The cause. The login page includes an IFRAME that gets blocked by Adblock Plus, probably because the URL looks similar to URLs often used for advertising.
In more detail, the login page includes an IFRAME sourced from a URL of the form
https://openid.stackexchange.com/affiliate/form?...&callback=http%3a%2f%2fsecurity.stackexchange.com%2fusers%2fauthenticate&.... This IFRAME is blocked by a rule from Fanboy's List (English) and EasyList (USA): namely, the
com/affiliate/$~object-subrequest rule. It appears that this rule blocks all access to any URL of the form
*://*.com/affiliate/*, under the assumption that these URLs typically indicate some form of advertising.
Solution. There's a simple method to fix the problem and make Stack Exchange login work again: add a custom exception to whitelist this particular URL:
From the Firefox menu, click on Tools >> Adblock Plus >> Open blockable items (or press Ctrl + Shift + V). This should open a small window showing all items on the current page, one row per item.
Start to log in on a Stack Exchange Site. On the page where the login process hangs, you should see one row with a big red X next to it showing the URL I listed above. Right-click on that row.
Add a custom Exception Rule. I selected my Exception Rule to apply to
openid.stackexchange.com/affiliate/*. Enable this rule. This will whitelist the frame that is used by the Stack Exchange login.
Test logging in again. Everything should now work.