After reading your question, I had a vague memory that this subject was indirectly touched upon in "On the Origin of Species", so I did some text searches (in this pdf version I found online). From what I can see, Darwin never used the technical term 'variance' (I don't know how old this use of the word is), but 'variability' is often used, both with regard to accumulated variation but also meaning variation between individuals within a population.
All-in-all, I could not find clear support for the idea that selection will decrease the amount of variation. However, there are number of passages where the subject of variation vs selection comes up (often indirectly), but, to me, these just as often (most often?) indicate that Darwin saw no risk that selection would reduce variation as the opposite. Here are the most interesting passages I could find (searching for combinations of 'variab*', 'selection', 'decrease' etc), with page references referring to the pdf version and a short comment below each quote:
A large number of individuals,
by giving a better chance for the appearance within
any given period of profitable variations, will compensate
for a lesser amount of variability in each individual,
and is, I believe, an extremely important
element of success.
(p. 44)
This is clearly discussing amounts of variation within species, where large numbers can compensate for a smaller amounts of standing variability, by increasing the chance of advantageous mutations.
Nothing can be effected, unless favourable variations
occur, and variation itself is apparently always a very
slow process. The process will often be greatly retarded
by free intercrossing. Many will exclaim that
these several causes are amply sufficient wholly to
stop the action of natural selection. I do not believe
so. On the other hand, I do believe that natural selection
will always act very slowly, often only at long
intervals of time, and generally on only a very few
of the inhabitants of the same region at the same time.
(p. 103)
Not directly related to selection vs variation, but deals with the rate of evolution vs amount of variation.
But what here more
especially concerns us is, that in our domestic animals
those points, which at the present time are undergoing
rapid change by continued selection, are
also eminently liable to variation. Look at the breeds
of the pigeon; see what a prodigious amount of difference
there is in the beak of the different tumblers,
in the beak and wattle of the different carriers, in the
carriage and tail of our fantails, &c., these being the
points now mainly attended to by English fanciers.
Even in the sub-breeds, as in the short-faced tumbler,
it is notoriously difficult to breed them nearly
to perfection, and frequently individuals are born
which depart widely from the standard. There may
be truly said to be a constant struggle going on between,
on the one hand, the tendency to reversion to
a less modified state, as well as an innate tendency
to further variability of all kinds, and, on the other
hand, the power of steady selection to keep the breed
true. In the long run selection gains the day, and we
do not expect to fail so far as to breed a bird as coarse
as a common tumbler from a good short-faced strain.
But as long as selection is rapidly going on, there
may always be expected to be much variability in
the structure undergoing modification.
(p. 141)
This is maybe the most interesting passage, which to me indicates that he saw little risk that selection could/would limit variability (see especially beginning and end).
For forms existing in larger
numbers will always have a better chance, within
any given period, of presenting further favourable
variations for natural selection to seize on, than will
the rarer forms which exist in lesser numbers. Hence,
the more common forms, in the race for life, will tend
to beat and supplant the less common forms, for
these will be more slowly modified and improved.
(P. 162)
Also on limitations for variation that selection can act on, and the value of large populations.
... firstly, because new varieties are
very slowly formed, for variation is a very slow process,
and natural selection can do nothing until
favourable variations chance to occur, and until a
place in the natural polity of the country can be better
filled by some modification of some one or more
of its inhabitants. And such new places will depend
on slow changes of climate, or on the occasional immigration
of new inhabitants, and, probably, in a still
more important degree, on some of the old inhabitants
becoming slowly modified, with the new forms
thus produced and the old ones acting and reacting
on each other.
(p. 163)
On the limiting nature of variation, but does not connect selection to decreased variation.
It cannot be asserted that organic beings
in a state of nature are subject to no variation; it cannot
be proved that the amount of variation in the
course of long ages is a limited quantity; no clear
distinction has been, or can be, drawn between spe-
cies and well-marked varieties.
(p. 425)
On variation over the long run.
These are the most interesting sections I found on the subject from my (quick and limited) searches. I hope you find them useful. However, this is only based on the first edition of "On the Origin of Species", and it is possible that it is is modified in later editions or that he touches upon this in his other writings. However, when looking at these particular passages in the sixth edition, I cannot see a major change in his thinking.
