Okay, got it figured out, sort of. Tapatalk support responded and asked me to see if I could login to the iPhone Forums website using a browser on my phone. Much to my surprise I was unable to do so, either from Chrome or Safari. I was able to access other websites from my phone browsers, and Tapatalk was able to connect to the forums if I used 4G instead of Wi-Fi. I was able to access the iPhone forums from a laptop connected on the same Wi-Fi. So I knew it wasn't a Tapatalk problem. Thinking back, I remembered that about a week ago I had made a security change on my router - I had disabled WPS. I reenabled it, and now everything is working again.
What's bothering me is that:
1) I don't know what disabling WPS would have to do with the connection issue.
2) I would have thought that if disabling WPS would stop me from connecting to the iPhones forums from my iPhone that it would also break other connections too. Remember that I had still been able to connect to the other forums that I access using Tapatalk.
3) I had been able to access the forums from a browser on my W-Fi connected laptop, so this is an issue specific to the iPhone's Wi-Fi connection, not any Wi-Fi connection.
Although I can live with the WPS disabled, I'm really curious as to what's really going on. Any ideas anyone?