I have some more information about [what I believe to be] this problem. We've had similar incidents from several clients, running kernels 3.16.{36,39} and 4.9 (jessie-backports). I think this rules out a client hardware issue.
The trigger (from the client's perspective) seems to be loss of contact with the NFS server. The incidents are almost always preceded by one or more nfs: server <name> not responding, still trying log entries. Sometimes there is a known server-side explanation (e.g., nfsd thread exhaustion), but not always. In any case, the effects persist well after communication with the server has recovered; "reboot -f" seems to be necessary for client recovery, as sync() also hangs indefinitely. Kernel stack traces on the client vary, as do the affected files and applications; the issue is by no means limited to Firefox or sqlite. If desired, I can submit a selection of stack traces (as one bug or as several). I'm looking for suggestions on how to debug this. I'm thinking of turning on logging with rpcdebug on the the most frequently affected clients, to better understand the trigger. Is there anything else I should be looking at?