TL;DR: Just keep the firmware up to date.
In this post I reported that my Azure S2S was no longer working. Now, troubleshooting 101: At time A it worked. At time B it stopped working. What happened in between those times? On my end, nothing. There were no changes made to the configuration of my Fortigate.
What I noticed at the time was that the link stayed up intermittently, then failed entirely.
Long story short, I upgraded from FortiOS v5.4.8,build1183 (GA) to 6.0 this weekend. S2S is working again.
So I have to assume that some changes occurred on the Azure side that stopped working with the older version of FortiOS (although I did go back and try several of the configurations I found on various sites, including the Fortigate cookbook). Changes that now work with 6.0 and no changes to my config.
Thanks to everyone who replied with suggestions.
EDIT: The delay in upgrading was that I bought my device on eBay. I got it registered with Fortinet, but had to buy a couple of back years of the Forticare contract. Still, money well spent.