An issue with fireproofing objects is that they often need to be encased (think of a fireproof safe) to protect them if the room they are in is on fire. So for a backup computer (or external HDD, etc) - how would the they receive electrical power, and how would they be ventilated to prevent overheating?
If you're serious about backups remember the following (can't remember the source):
"Don't have a backup plan, have a recovery plan"
Simply backuping up to a removable HDD or DVD and placing it in a suitable fireproof safe (not all fireproof safes are rated to protect electronic media and/or electronic devices!) might be sufficent to protect your data, but doesn't hit your "forget" about it criterion.
That said, the danger with a setup-and-forget backup system is you might not notice when it isn't working, which can be a serious problem.
The best fireproofing for anything (backups included) is for it to be somewhere else entirely.
So, an External HDD / CD / DVD that you leave somewhere safe (work/with a family member/etc) would work, or you could maybe consider looking at an online backup solution.