Why should I not keep things producing static electricity near my computer cabinet?

Sorry, but your Dad is probably being a bit too careful.

You're never going to kill a computer with static by touching its keyboard, mouse, or case.

Static damage does not affect well designed, modern electronics devices (including computers). Perhaps in the 80's - but not now.

Since my Degree in Electronics Engineering, I've killed probably 30 Integrated Circuit (IC) chips by unintentionally zapping them with rogue static charge on my fingers. However, once chips are mounted on a well designed board, the ground plane helps to make the entire circuit more robust, and the case helps to protect the entire product. In addition, IC manufacturers have been getting steadily better since the 70s at adding internal static protection circuits to avoid this very problem.

Bottom line:

• You can ignore static damage if you're a home computer user. Modern electronic devices are so well designed that its not a problem any more (although it was up to the 80s).
• You must take care to avoid static damage if you're an electronics manufacturer, home electronics hobbyist (e.g. soldering ICs to a board), are installing computer parts (especially RAM or hard drives), or you are doing something super critical like installing a pace maker.

Generic eHow articles and How to Prevent Static Electricity Damage to Electronic Devices.

Static electricity can disable or even destroy your mouse, keyboard, printer, fax machine, even your computer's circuit boards (and all your data along with them).

^{Paranoia Alert: this sentence is probably an exaggeration -- check details by Gravitas.}

Another thing to remember is that the usual method of cleaning and wiping surfaces is to rub with a dry cloth. The usual kind of surface (plastic, glass) and material of wipe in use can promote static charging very quickly.
^{You can wipe a surface clean and end up wiping the electronics under it to death -- ok, that's an exaggeration :-)}

Update: I agree with Gravitas on the count of 'well designed and modern devices'.
The fear of static discharge affecting electronics should be limited to conditions when you have opened up such hardware and would get your fingers or other tools close to the components.

I almost deleted my answer in retrospect. But, I guess its good to keep it around for context.

Static can be discharge through any electronic devices (well, any conducting object).

When it does discharge it does so at a large voltage, and it's this high voltage that can potentially cause damage to sensitive components (such as most of the innards of your computer).

If you think this seems a little unlikely that it can cause damage, remember that lightening (as in the storm based variety) is the same discharge effect - just on a larger scale.