What is Hinduism's stance on Buddhism?
Does Hinduism declare that Buddhism is wrong?
If yes, what is it about Buddhism that Hinduism declares to be wrong?
If no, why not?
Buddha is accepted as a lila incarnation of vishnu.
IOW Lord Buddha appears periodically to fulfill the same purpose although details may differ slightly between subsequent incarnations.
For instance here is a passage in reference to a incarnation of Buddha from a previous universal creation.
SB 2.7.37
When the atheists, after being well versed in the Vedic scientific knowledge, annihilate inhabitants of different planets, flying unseen in the sky on well-built rockets prepared by the great scientist Maya, the Lord will bewilder their minds by dressing Himself attractively as Buddha and will preach on subreligious principles.
The purpose of Buddha however is the same in all times (at the onset of kali yuga) and situations (namely a misuse of vedic knowledge by an dominant atheist class and remedying the situation with a philosophy of ahimsa - non-violence) - for instance here is a passage in reference to this universe.
Dasa avatar stotra 4 - O Kesava! O Lord of the universe! O Lord Hari, who have assumed the form of Buddha! All glories to You! O Buddha of compassionate heart, you decry the slaughtering of poor animals performed according to the rules of Vedic sacrifice.
In short, brahmanas were performing vedic sacrifices simply for the sake of eating slaughtered animals (aka atheism) , so Buddha incarnated and established the rejection of vedic authority and the establishing his own authority (namely non-violence with a tinge of impersonal liberation as perfectional)
Buddha turns up many times in various prayers through out vedic literature
SB 8.3.12:
I offer my respectful obeisances to Lord Vāsudeva, who is all-pervading, to the Lord's fierce form as Lord Nṛsiḿhadeva, to the Lord's form as an animal [Lord Varāhadeva], to Lord Dattātreya, who preached impersonalism, to Lord Buddha, and to all the other incarnations. I offer my respectful obeisances unto the Lord, who has no material qualities but who accepts the three qualities goodness, passion and ignorance within this material world. I also offer my respectful obeisances unto the impersonal Brahman effulgence.
What is interesting however is how Buddha is seen as playing a role in a religious continuum - what overtook Buddhism (and eventually pushed it to the perimeters that it is now famous for) was sankharacarya's mayavada philosophy (which is basically just establishing the values of buddhism - namely impersonal liberation according to vedic authority). IOW sankharacarya re-established the authority of the vedas after buddha dismissed them (sometimes even scholars refer to mayavada philosophy as "veiled Buddhism"). Basically as the continuum progresses each participant is seen as offering something beneficial - in the case of buddhism it is the Amara Kosa, or sanskrit dictionary, for establishing authoritative explanations that all parties involved agree on.
Sometimes there may be vedic references that philosophically condemn buddhism simply because it is a branch of atheism, although Buddha himself is understood as an incarnation of Visnu. As such, to really get in the spirit of following Buddha requires that one is an atheist, so his appearance day, etc won't be celebrated in any mandir.
Basically in vedic discussion religious paradigms are argued in terms of their linear position in terms of philosophy - its not that buddhists are "wrong" or go to hell, its just that the ultimate result it offers pales in comparison to whats on offer. Basically the faults lie in the explanation of the phenomenal world and self and also Buddhistic nirvana being attainable as an eternal goal.
(on a side note - In 1959, subsequent to the then ongoing Chinese occupation of Tibet, the 14th Dalai Lama sought refuge within India. The then Indian Prime Minister, Jawaharlal Nehru, was instrumental in granting safe refuge to the Dalai Lama and his fellow Tibetans. The Dalai Lama has since lived in exile in Dharamsala, in the state of Himachal Pradesh in northern India, where the Central Tibetan Administration (the Tibetan government-in-exile) is also established. Tibetan refugees have constructed and opened many schools and Buddhist temples in Dharamsala. - wiki -
the simple fact that the tibetan buddhists were granted shelter in India is indicative of the traditional vedic social culture of india)
If you want to talk of the "wrongness" of Buddhism, then this is probably more down that alley
You practice Buddha if you appreciate him. You give up everything like Buddha and meditate. But that you will not do. Then what is the talking of, useless talking about this? Do something. Either you believe Buddha or Jesus Christ or Krishna. Do something. Don't talk simply. Lord Buddha is very nice. He gave up his kingdom in youthful life. He was prince. He thought, "It is all nonsense. Let me meditate." Do like that. That is the disease. We won't do anything. We talk much of this, that, this, that. Do anything, but do it perfectly. "Jack of all trade, master of none." That is not good. Be master of something. It doesn't differ. Either you follow Lord Jesus Christ or Lord Buddha or Krishna, it doesn't matter much. But do it perfectly. That is our request.
- A.C. Bhaktivedanta Swami Prabhupada
once again, theory only becomes fruitful with practice