Yes, it will fill up. Consider implementing a cron job that will delete old files after a while.
Something like this should do the trick:
/usr/bin/find /tmp/mydata -mtime +1 -exec rm -f {} \;
This will delete files that have a modification time that's more than a day old.
Or as a crontab entry:
# run five minutes after midnight, every day
5 0 * * * /usr/bin/find /tmp/mydata -mtime +1 -exec rm -f {} \;
where /tmp/mydata is a subdirectory where your application stores its temporary files. (Simply deleting old files under /tmp would be a very bad idea, as someone else pointed out here.)
Look at the crontab and find man pages for the details. Don't go running scripts that delete files on your filesystem without understanding all the details - that's how bad things happen to good servers. :)
Of course, if you can just modify your application to delete temporary files when it's done with them, that would be a far better solution, generally.