Does savings always equal investment? Are there any cases where savings could be removed from the economic system altogether and break the balance? Assuming a closed economy, does savings always have to equal investment?
If an individual produces a certain amount of a resource and saves it, that becomes part of the savings of the economy. However, if the resource is subsequently destroyed, it is then removed from the economy. In that case, the savings have increased but the investment hasn’t?
Does destroying a resource count towards investment? Is that an edge case we don’t consider at all? Is this any different from the way investment takes resources away from an individual and puts it into an economy?
Does savings always equal investment?

If we are talking about national accounts then yes. Output of closed economy is given by:
$$Y = C+I+G \tag{1} $$
Private saving $S_p$ is by definition is disposable that is not consumed (see further explanation in Blanchard et al. Macroeconomics an European Perspective pp 55) so:
$$S_p= Y- T-C \tag{2}$$
Public saving $S_g$ is by definition equal to difference between taxes $T$ and  gov spending $G$ (see ibid) so we have:
$$S_g= T-G \tag{3}$$
Substituting 3 into 2 (by solving for T) and then into 1 (by solving for Y) gives us:
$$S_p+S_g + G + C= C+I+G \tag{4} $$
Now just by simple algebra we can see $G$ and $C$ are eliminated so we are left with:
$$S_p+S_g =I$$
So total saving (public and private), must be at all times equal to investment. The above is pure accounting identity, it simply must hold the same way as it is not possible that in some company assets are not equal to liabilities and equity.

However, if the resource is subsequently destroyed, it is then removed from the economy. In that case, the savings have increased but the investment hasn’t?

No. You need to realize that $C, I, G, Y  $ or $S$ must be defined over some time period like a day, month, year, decade and so on. Let us assume that we are talking about yearly period for a simplicity sake. For example suppose that in 2010 goods worth of 1000e were produced, people wanted to save 200e but in the same 2010 100e  were destroyed (gov is assumed away). Well the GDP gets calculated at the end of the time period so if we would look inside the time period before destruction GDP would look like $Y=1000$, $Y-C=800$ $S=I=200$ but after destruction it would look like  $Y=900$, $Y-C=800$ and $S=I=100$. You see that it would all perfectly balance and overall what is reported at the end of the year is  $Y=900$, $Y-C=800$ and $S=I=100$.

Does destroying a resource count towards investment?

No, but it will no longer count toward output as well. Investment will be lower but output for the same period will be lower by equal amount making it all equal.
To give you an intuition, you should realize that $Y$ output/income must always be equal all spending $C+I+G$ (for closed economy), if you lower spending on  $I$ (because part of it is destroyed and thus no longer counts), $Y$ has to drop as well.  What you are saving in real terms is not money but output -that is what matters, you are consuming output, investing output (which is equivalent to saving output as proven above) or government is consuming output - it is all just breakdown of what happens with output Y, saving part of it does not make it no longer output.

Is this any different from the way investment takes resources away from an individual and puts it into an economy?

Yes of course. Investment is basis for capital formation which in turn expands the societies production possibility frontier (see discussion of this in ibid ch 14).
Destruction of those resources will lower the total investment, hence ceteris paribus negatively affect future output (however, note here everything is expressed in real terms not nominal terms i.e. destroying money would not have the same effect as that would just change nominal value). However, as explained above it also lowers present income so national accounts perfectly balance each other.