No databases will go offline. It wil generate error level 17 messages stating "object xx can't be created in filegroup yy for database TempDB" and the transaction(s) will be cancelled and rolled back.
Unless it happens during a restart. If tempdb does not have enough space to growth, SQL won't start.
If TempDB cannot be recreated, SQL will not start. That would required that someone shuts down SQL, deletes the tempDB files and then does something to fill the disk before restarting SQL. Otherwise the TempDB files will already exist at the time of startup (and they're not grown during the startup process, they're set back to their initial size)
Microsoft Certified Master: SQL Server, MVP, M.Sc (Comp Sci)SQL In The Wild
: Discussions on DB performance with occasional diversions into recoverability
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