Fortinet: Is there any equivalent of the ASA's packet-tracer command?

On the Fortigate you actually don't have command with capability to generate a dummy packet like on your cisco ASA. But the closest utility will be "diagnose debug flow" commands. The difference is that, with fortigate you need real traffic traversing through the firewall.

Below are the complete commands that you need to execute:

diagnose debug reset
diagnose debug flow filter addr <source OR destination IP address>
diagnose debug flow show console enable
diagnose debug flow show function enable
diagnose debug flow trace start <number of entries you want to view. e.g. 100>
diagnose debug enable