What happens when a file size limit is reached? I have a SQL Server 2008 R2 Standard Edition database for which I have configured the ROWS (.mdf) database file with autogrowth ON and set to max size to 10GB.

After a while, I get an alert that my tempdb log has filled a drive. I figure that my database's size is stuck on the limit at 9990MB or something, unable to extend any further.

Are those two things possibly related? Would the database, being unable to expand, store its transactions in tempdb?

What happens when a database ROWS file gets filled and someone keeps adding data to it?

The data was coming from SQL Server performance data monitoring:
No, the two are unrelated. 

TempDB is used for explicit temp tables, but also for sorting and database snapshots and lots of other internal things. Was there some other very large transaction going on when the TempDB filled up?

If a transaction needs more data space (or log space, or TempDB space) but the file is set to not grow (or there is insufficient space on disk), the transaction will fail and roll back. So no new data will be added.