Shankara's description of Buddhists in his time
Among them there are three schools: some are Sarvastivadins (divided
into the Sautantrikas and Vaibhasikas, believing respectively in the
inferential and perceptual existence of all things); some are
Vijnanavadins (or Yogacaras, believing in the existence of
consciousness or ideas alone); while others are Sarvasunyavadins (or
Madhyamikas, denying the existence of everything), (i.e. they are
realists, idealists and nihilists respectively.
Brahma Sutra Bhasya of Sri Shankaracharya, Chapter II, Topic 4: Refutation of Buddhist Realists, translated by Swami Gambhirananda
A short summary of Shankara's treatment of Buddhism
There are realistic as well as idealistic schools in Buddhism. All
things are aggregates, according to Buddhism; there is nothing like
substantiality. For the realistic schools, there are two kinds of
aggregates, the internal and the external. But, consistent with the
other Buddhist doctrine of momentariness, how aggregation can take
place at all passes one's understanding. There is a processor of
successive moments; but how are the moments related? What connection
is there between what precedes and what succeeds? These questions
remain unsolved. For the Buddhist idealist, there is no extra-mental
reality; ideas are things; what is real is a series of momentary
ideas. This view is also untenable. The appearance of ideas is sought
to be explained as brought about by residual impression. But, how can
there be residual impression, if there are no external things. So says
Badarayana, the Buddhist view is totally unintelligible
(Sarvatha-anupapattih: Brahma Sutra II.II.32).
Foreword to Brahma Sutra Bhasya of Sri Shankaracharya by T. M. P. Mahadevan
Here are some detailed arguments given by Shankara against Buddhism. He dismisses Sunyavadins as nihilists.
The major problem is with the absence of Atman in Buddhist thought. The lack of an Atman would create the problem of instability regarding the personality of an individual. Shankara attacks the philosophical position of the various Buddhist schools and shows the validity of the Atman principle. I will quote Sankara's Brahma Sutra Bhasya II.ii.31 where he attacks the Buddhist Idealists (Vijnanavada), Buddhist Realists (Sarvastitvadins) and Buddhist Nihilists (Sarvasunyavadins or Madhyamikas) to give a flavor of his argument:
As for the ego-consciousness that is assumed to be the abode of
disposition (or tendency), that too has no stable form, since you
postulate its momentariness like sense-perception. Hence it cannot be
the abode of tendencies. For unless there be some principle running
through everything and abiding through all the three periods of time
or some unchanging witness of all, there can be no human dealing
involving remembrance, recognition, etc, which are contingent on past
impressions that are stored up in conformity with environment, time
and causation. If the ego-consciousness be (assumed to be) unchanging
by nature, your doctrine (of momentariness) will be set at naught.
Moreover since the theory of momentariness is upheld equally in
Vijnanavada, all the defects arising from momentariness that were
levelled (by us) against the theory of these (Buddhists) who believe
in the existence of (momentary) external things, viz those shown under
the aphorisms starting from, "And because the earlier is negated when
the later emerges" (II.ii.20) are to be remembered in this context as
well. Thus are refuted both these Buddhist points of view - of both
those who believe in external things and those who believe in
(subjective) consciousness). As for the view of the absolute nihilist,
no attempt is made for its refutation since it is opposed to all means
of valid knowledge. For human behaviour, conforming as it does to all
right means of valid knowledge, cannot be denied so long as a
different order of reality is not realized; for unless there be an
exception, the general rule prevails.
Brahma Sutra Bhasya of Sri Shankaracharya II.ii.31
What does Shankara mean in the passage given above?
You notice how he brings in the Atman theory (first bolded sentence) indirectly. What does Shankara mean? He is attacking Buddhists who think of the 'I' sense in the following manner: I .... I.... I (where the 'I' sense does not exist during the dotted time period). What Sankara is arguing is that how do these Buddhists know that the series is not I1...I2...I3 etc where I1, I2, I3 are three different ego-consciousnesses? How can there be a stable personality which remembers a unique past or recognises old friends if the ego is unstable? In fact it is these Buddhists who need an unchanging principle (the Atman) that witnesses everything for all time (i.e. even during the gaps in ego-consciousness). Only if this Atman exists can Buddhists avoid problems regarding stability of personality. Otherwise a person who is Rama at one moment will consider himself Lakshmana in the next moment after the ego comes back. If these Buddhists now say that the ego-consciousness is stable and not momentary in order to save themselves from this conundrum then they have refuted themselves. Sankara then goes on to say that he can give a similar argument refuting Buddhist realists who regard the external world to exist momentarily. Even in this case of a momentary external world you will need an Atman (an unchanging witness for all time) to give stability to our perception. So in either case you need the Atman principle to make sense of our experience.
Shankara thinks that the Madhyamikas deny the existence of everything which is untenable.