Try clearing site cookies and disabling Tracking Protection for the site.

Does it still happen in Troubleshoot Mode?

Thank you for the technical advice but I was looking for a simple answer that would not change vital settings and require something like constant cookie clearing. There must be untold thousands of people who have experienced this problem, so surely Ancestry and Firefox have done something about it.

If you use extensions ("3-bar" menu button or Tools -> Add-ons -> Extensions) that can block content (Adblock Plus, NoScript, DuckDuckGo PE, Disconnect, Ghostery, Privacy Badger, uBlock Origin), always make sure such extensions do not block content.

• make sure your extensions and filters are updated to the latest version
• https://support.mozilla.org/en-US/kb/troubleshoot-extensions-themes-to-fix-problems

You can check for issues with Enhanced Tracking Protection. Firefox shows a purple shield instead of a gray shield at the left end of the location/address bar in case Enhanced Tracking Protection is blocking content.

• click the shield icon for more detail and possibly disable the protection

You can check the Web Console for relevant-looking messages like blocked content.

• https://support.mozilla.org/en-US/kb/enhanced-tracking-protection-firefox-desktop
• https://support.mozilla.org/en-US/kb/total-cookie-protection-and-website-breakage-faq
• https://support.mozilla.org/en-US/kb/trackers-and-scripts-firefox-blocks-enhanced-track

It is possible that security software is causing the problem. Try to disable security software temporarily to see if that makes a difference.

As I have said, surely there is some way where two major, mainline programs will work together without extensive fiddling around with settings. I don't want to go into Firefox and start changing settings that might affect my other usage. I have already tried disabling my ad blocker wrt Ancestry and that did not help at all.

Does it still happen in a new profile? An easy way to test a new profile is to install Developer Edition and see if it happens there or refresh your existing profile. Make sure you stay signed out of your sync account when testing.