Does anyone know what effect slightly underdriving an MH lamp would be, with regards to the probability of exploding at EOL? Since the arctube would be running cooler, would it lower the chance of explosion?

Probably would, but you might get an opposite effect too : If due to lower temperature some halide is only liquid instead of vaporized and pools in a single spot of the arc tube, it may react with the wall and compromise it at that spot

How "slightly"? Probably nothing if you run a 100watt standard MH lamp with a 70w ballast. Might be bluer, as the last halide that gets driven to max is the sodium (as in pressure broadening of spectral Na lines). I can tell you that I plan to find out exactly what happens in such a situation (spectra wise) as well as MH & HPS missmatching. I've seen some unusual colors in typical HPS mini wallpacks that I think I know what's going on, or at least, I'll narrow it down... In due time.

Depends on a few factors, how much do you consider "slightly"?, What is the lamp chemistry and operating position?, and whats the ballast situation. Ive seen plenty of both good and bad outcomes from underdriving MH lamps, and not one includes violent failure of the lamp. In the dozens of misapplications ive seen where a high wattage lamp is operated on a lower wattage ballast, the lamp always seems to win and the ballast is either cooked, or the capacitor fails. Universal burn lamps operated horizontally while on the incorrect (lower wattage) ballast usually cycle right away, or in short time. If anything youll have a noticeable flicker. Base up burn lamps seem to fare better and actually operate somewhat stable on lower power, albeit at a much cooler color temp. But again you run the risk of overheating the ballast.

Now, that's only regarding standard magnetic ballasts that are designed to output a fixed voltage and current...

Dimmable electronic, and dimming-capable magnetic are a completely different ball game. Both of these types of ballasts ive seen can successfully operate a stable arc down to 50% rated power, regardless of lamp operating position, and with seemingly no negative affects to the lamp or ballasts. Yes, the color temp does still increase with power decrease, but its still the same smooth operation compared to full power, also from what ive seen, the lamps actually DO last longer on these systems compared to running at full rated power throughout life.

If you are interested in dimming your MH lamps I recommend investing in a dimmable ballast. Ive seen way too many situations of failed ballasts due to lamp mismatch, even a 100w (M90) running on a 70w ballast. Though most of the time ive came across 250w lamps on 175w ballasts, which is likely due to the fact that those two lamps are nearly identical physically, and most people have the habit of just physically matching their spent lamps to new lamps at hardware stores. 

"Matching their spent lamps to new lamps at hardware store." The 175 and 250w metal halide lamps are almost identical physically, as well as aforementioned wattages in Mercury vapor lamps. I also see where people exchange lamps at stores to get a free one when in reality, they slip the eol one in the new sleeve and use the new One at home, effectively getting money back or a free lamp.

lol, when it comes to fluorescent, most people seem to get it, I rarely come across mismatched linear fluorescents. But when it comes to HID, all common sense seems to go out the window. "If it looks da same, then its da same bulb, eh". And yes, I could tell you stories about people trying to return "defective" lamps they just purchased to the electrical supply house. As if they don't think the staff can easily discern a spent arctube from a new one, lol sometimes they don't even get the brand right.

For fully coated lamps, it's very hard to tell, unless you shine a light up to it. And I've seen mismatched fluorescents: a f18t8 and f20t12 operating on a series magnetic rapid start ballast. That's probably why I only got it to glow in the ends, even with proper lamps.

With fluorescent is in most cases pretty impossible to mismatch the lamp and ballast, just because the wrong lamp does not fit at all. Of course, some combinations do fit (in Europe I can not recall any such combination), unfortunately it is the case of the most common North American commercial size of F40T12 vs F32T8. But there the lamps are supposed to be replaced by dedicated worker (as it usually includes working at heights, many times the ocveral lantern inspection is supposed to happen during the relamping,...) and there I would expect this worker to know what he should do and what to take care of.

The risk of arctube explosion may be with some designs and conditions more likely due to the stresses the design does not count on (e.g. more frequent cycling), but definitely it would not be as violent as when it happens at full power, so way more likely contained by the outer.

There is quite significant difference between underdriving and dimming:
With "dimming" you use ballast designed to operate the lamps at reduced power, but rated for that lamp.
That means regardless of the setting, the first 15 minutes after power ON the ballast system drives the lamp at full power, only then it eventually reduces the power to the desired level. This 15minutes of full power is there quite essential to set all the parts of the lamp into the required operating mode (properly warm up the electrodes, evaporate and mix the fill, let the halogen cycle to clean the burner,...). After this conditionning it is then possible to reduce the power really deeply without the detrimental effect on the lamp. Plus the ballast is designed to handle the shifted properties of the lamp, so it will not get harmed by e.g. lower arc voltage or so.

A ballast just underdriving the lamp does not precondition the lamps and that means the lamp may not start properly, e.g. stay in the low pressure mode (deadly for most ballasts), the fill components may separate because only part of it is evaporated each time, the electrodes may run at too low temperature, so sputter,... Plus the ballast may not be designed to work with the reduced arc voltage (mainly when the unsaturated vapor lamp does not warms up into the unsaturated vapor mode - that could be the case even for MH's, which operate with some components still in saturated mode, as the mercury component is usually supposed to be unsaturated), so become overloaded and so the ballast failures.

What I find confusing for the general public (or anyone not familiar with HID) is the labels manufacturers place in fixtures. With an incandescent fixture, if the label reads 'maximum 100 watts', I can use any incandescent lamp I want provided it is a maximum of 100 watts. If I want to use a 7 watt night light lamp, it will work just as well as the 100 watt. With HID, you need to match the lamp with the ballast (here in N America, ANSI codes). However, the manufacturers have the habit of stating, on a 100 watt MH fixture for example, 'use only M90 lamp 100 watts maximum'. The uninformed will see that and use a lower wattage lamp since it might be the only one readily available and it would work with the incandescent lamp logic. The remedy would be for the manufacturer to stop using the word 'maximum'. I guess it wouldn't stop every lamp and ballast mismatch, but it could help somewhat.

I saw my yardblaster said  "use max 175w Mercury vapor lamps". I could imagine why the ballast was acting up.

It is because of the electrical Code being quite in delay (already about 40 years) behind the technology...
There is quite a conflict between what the system really needs and the safety code: Practically all codes around the world require on consumer light fixtures to "state the maximum wattage of the light source". When this label is missing, the item is assumed (mainly by lawyers) as violating the code, so "dangerous". Even when it is a LED lantern with LED's designed as integral part of the fixture (so not replaceable).
So even a F8T5 fixture has to bear the label saying "Maximum F13T5", even though you just can not fit any other lamp and actually the lower wattage lamp, when connected, would be likely to cause ballast fire (speaking about preheat fixture)...

In some of the newer fixtures I have bought that are HID the nameplate in the fixture and ballast says "Use ONLY with 100W MH M90 lamp"Then it goes on to say "Other lamps may cause fire and electric shock hazard"This was in a 100W MH wall pacKs I boght from Lowes or HD.In one of these light with a blown ignitor a 100W mercury lamps runs just fine.Rescued that light from the junkbox!

I prefer the word 'only' to 'maximum' when talking about discharge lamps. Makes much more sense for the uninitiated.

Wow, sorry everyone. I got no email that anyone ever replied to this.

How "slightly"?

I didn't have a concrete value. I suppose down to 50% rated current, or thereabouts.

Anyway, the purpose of my asking is I'd like to run my one MH lamp more often, just to look at it or take pictures or whatever, but it's an enclosed-rated lamp, and I have nothing to run it inside that would simultaneously allow me to view the lamp, and contain an explosion. So I was just wondering if underdriving a lamp would reduce the chance of explosion, and thus make it relatively safe to run in the open.

Here's a related question - it's my understanding that an MH arctube is at (or only a little bit above) atmospheric pressure when cold. Does the pressure increase as soon as the arc strike, or only once the halide salts begin vaporizing and/or the arctube begins heating up? Another way to put it is, could an MH lamp theoretically explode the instant it ignites, even while still cool?

It depend, what would be the cause for the explosion.
When cold, there is nearly a vacuum inside of the arctube.
But with a shorted ballast, when the arctube becomes practically directly connected to the mains, after ignition the arc may fatten so quickly, the arctube may explode. The forces do not come only from the pressure, but are to big extend of an electromagnetic nature, so do not need the arctube to warmup first.

Normally (no defect in the ballast or so) the pressure starts to rise steeply once the mercury vaporises. In fact that is the main pressurizing element in the arctube and it remains fully evaporated even with quite deep dimming.

But the problem would be, to prevent the lamp from operating in some ill mode and to prevent fast degradation, even when dimmed, the lamps are required to be warmed up and stabilized for at least 15 minutes at full power (during that the complete fill is evaporated, so mixed throughly, the halogen cycle may clean the arctube and few other "restoration" mechanisms take place). And this 15 minutes (mainly the warmup phase) is, when the explosion is most likely to happen.
And not meeting the 15minutes full power requirements means the sputtering will blacken the arctube, the fill components will get separated and few other effects will degrade the lamp.

The 15minutes comes from maker's recommendation for dimmed system, the actual lamp requirement may be shorter, but the point is still the same: The lamp has to fully warm up and the internal dose fully (as rated for full power) evaporate and then burn at the full power for some minutes, so the most critical time section (for the eventual explosion) is still there...