Hello Eric

Welcome to our RO forum!

Yes, it can harm the mask of the picture tube permanently, even when the TV is off.
I looked up this here in 15 seconds:
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A few more inches between monitor and speakers would do the same trick, though.

mu metal is the best magnetic shield i have used.
[[url=http://[/URL]="http://www.mumetal…"]About MuShield high permeability magnetic shielding material[/]="http://www.mumetal…"]About MuShield high permeability magnetic shielding material[/]

If you use permalloy or Mu Metal, keep in mind that you must try not to deform it too much or to hammer it.
The material looses its permeability quite dramatically, like μr=50.000 and when mechanically stressed only μr=150 is left.

Mumetal is undoubtedly the best material for this, but is both expensive and probably over the top for what you need. I would try a couple of sheets of plated mild steel ("tinplate") and experiment with those postioned between the monitor and the display. Test the suitability of the sheet by checking that a magnet will stick to it. Note that any alloy-based material is of no use for magnetic field screening.

Most CRT displays have either auto-degauss on power on or a degauss option in their function controls, so you are unlikely to cause permanent disruption to the monitor operation.

Build-in Degaussing safes you only to a certain degree. I advice not to rely on this automated de-magnetising.
I have fixed a bunch of picture tubes with an hand-held degausser because the damage through permanent magnetic fields
had bend the mask beyond of what the build-in coil could fix. Most of those defects were caused by speakers.

Something like here for 30 bucks might work for you ( with some search you'll find it cheaper)
http://www.orgoneshop.com/?open=71&showgroup=71&set=0&product=389&lang=en

Pardon me if I'm stepping out of line here but a Zenith System 3? Is that monitor really that good? A 15" LCD can be had for $100 that has as good if not better picture quality of a comparible CRT. I don't know. Am I missing something? I just tend to associate CRT monitors with problems in general. Noise being the most prominent one. I'm not claiming anything here. If I'm way off the path, please educate me. I've proven myself stupid many times before.