There is evidence, in fact a lot of evidence. Look at the lower text of the Sanaa manuscript for example.
There is a study by Dr. Kieth Small entitled "Textual Criticism and Qur'an Manuscripts" which does a comparison of the popular modern Quranic text to 22 other copies of the Quran including many of the earliest manuscripts we possess as well as several medieval manuscripts and one modern edition of the Warsh text used by many Muslims in North Africa and Yemen. He focused his work specifically on Surah 14:35-41. In just these seven verses, he found nearly 350 variants among all the manuscripts he examined.
Moreover, The very story of the Quran's collection reports the addition of at least one verse. In Sahih Al-Bukhari (Volume 6, book 61, number 509), it is said that after Muhammad's death there was a battle in which many people who had memorized the Quran had died. The close companions of Muhammad who were leading the Muslim community were concerned that portions of the Quran could be entirely lost if they didn't collect all of the Quran into one written volume. Abu Bakr sent a man to gather all of the Quran together, saying:
"you should search for (the fragmentary scripts of) the Qur'an and collect it in one book." He began the project of "collecting it from (what was written on) palmed stalks, thin white stones and also from the men who knew it by heart, till I found the last Verse of Surat At-Tauba (Repentance) with Abi Khuzaima Al-Ansari, and I did not find it with anybody other than him."
From this work, one official copy of the Quran was produced and remained with Abu Bakr until he died.
The very next Hadith in the collection (Volume 6, Book 61, Number 510) explains that by the time of the third Caliph, Usman, Muslims were reciting the Quran differently from one another so significantly that it threatened to tear the Muslim community apart. Usman sent for the full copy of the Quran that Abu Bakr had made, and also sent for all the written fragments of Quranic material throughout the land and commissioned a yet another project of collecting the Quran. He ordered the production of a new authoritative edition of the Quran, a copy of which was sent to each province. All the other Quranic material was burned. The hadith ends by noting that a verse had been left out of Abu Bakr's edition that was found and added to Usman's collection.
Once again, this verse was found only with Khuzaima bin Al-Ansari, the same person who had remembered a verse no one else knew the first time the Quran was collected.
Apart from the additional verses added. Hadith also provides evidence of missing verses.
In Sahih Muslim, Book 017, Number 4194, Umar is quoted as saying:
"Verily Allah sent Muhammad (may peace be upon him) with truth and He sent down the Book upon him, and the verse of stoning was included in what was sent down to him. We recited it, retained it in our memory and understood it."
In Sahih Al-Bukhari Volume 8, Book 82, Number 816, Umar is quoted as saying
"I am afraid that after a long time has passed, people may say, 'We do not find the Verses of the Rajam (stoning to death) in the Holy Book,' and consequently they may go astray by leaving an obligation that Allah has revealed."
A'isha, is also said to have reported:
"The Verse of stoning and of breastfeeding an adult ten times was revealed, and the paper was with me under my pillow. When the Messenger of Allah died, we were preoccupied with his death, and a tame sheep came in and ate it," (Sunan Ibn Majah, Volume 3, Book 9, Number 1944).
Moreover, we also find evidence that some of the early reciters that were considered to be great were also either forgetful or reciting different version of the Qur'an.
Sahih Al-Bukhari Volume 5, Book 57, Number 103 reads:
"Abdullah (bin Mas'ud) was mentioned before 'Abdullah bin 'Amr. The latter said, 'That is a man I continue to love because I heard Allah's Apostle saying, ' Learn the recitation of the Qur'an from (any of these) four persons: 'Abdullah bin Masud, Salim the freed slave of Abu Hudhaifa, Ubai bin Kab, and Muadh bin Jabal.' I do not remember whether he mentioned Ubai first or Muadh."
This same Abdullah bin Masud is also reported in Sahih Al-Bukhari as having said:
"By Allah other than Whom none has the right to be worshiped! There is no Surah revealed in Allah's Book but I know at what place it was revealed; and there is no Verse revealed in Allah's Book but I know about whom."
We cannot question, then, that Muslim tradition holds Masud as a particularly complete and trustworthy source for the Quran. Yet Muslim tradition is filled with hosts of examples of where Masud's readings were different from the standard text. In fact, some sources say his version of the Qur'an was three Surah's shorter.
Nonetheless, Masud criticized the version of the Qur'an compiled by Usman. He said
"The people have been guilty of deceit in the reading of the Qur'an. I like it better to read according to the recitation of him (Prophet) whom I love more than that of Zayd Ibn Thabit," (Ibn Sa'd, Kitab al-Tabaqat al-Kabir, Vol. 2, p.444).
He also accused Usman's scribes of adding three extra suras (1, 113 and 114) that had never been part of the original, and of making many other small changes to the text.
Ubai bin Kab is listed above alongside Masud as one of the great reciters of the Quran commended by Muhammad himself. Sahih Al-Bukhari Volume 6, Book 61, Number 527 records about Ubai bin Kab:
"Umar said Ubai was the best of us in the recitation (of the Qur'an) yet we leave some of what he recites."
Whereas Ibn Mas'ud omitted three surahs (1, 113 and 114) from his Qur'an mashaf (codex), Ubay ibn Ka'b had 116 surahs in his, including two extra short surahs, al-Hafd (the Haste) and al-Khal' (the Separation), which he placed between what are surahs 103 and 104 in Usman's Qur'an
Abu Musa al-Ash'ari, one of the early authorities on the Qur'an text and a companion of Muhammad, claimed a surah which resembled at-Tawba (also known as Bara'at) in length and severity was forgotten and lost, but included a passage on the greed of man, which is not in today's Qur'an. (Sahih Muslim Book 5, Hadith 2286). Even ibn Abbas was unsure whether it was part of the Qur'an or not (Sahih Bukhari Volume 8, Book 76, Hadith 445) Ubai said that it was considered as a saying from the Qur'an for a while during Muhammad's lifetime (Sahih Bukhari Volume 8, Book 76, Hadith 446). Al-Suyuti records the recollection by Abu Waqid al-Laithii of the occasion when the lost passage about the valleys was revealed. He says that Muhammad claimed it as a revelation from Allah, just like when he received other revelations.
(I would like to add here that the above also means that the whole bring a Surah like this challenge was already met back in the time of the Prophet and the caliphs as there were non-Qur'anic surahs and verses that sounded very much like those of the Qur'an. Surah al-Hafd and Surah al-khal', and the verses about Adam and the valleys sounded so Qur'anic that they were at one time believed to be so by speakers of 7th century Arabic, Sahabah no less. Those who claim that these were once part of the Qur'an and later abrogated, or that Al-Hafd and Al-Khal' were du'as given to Muhammad by Jibril need to explain why they were abrogated when there is no obvious reason, or why Allah allowed confusion to arise about the status of the latter two when they were recorded in the mashafs of three companions.)
Furthermore, people claim that early muslims had memorized the Qur'an which has ensured its authenticity. Yet, we find from Hadith that even Muhammad himself used to forget verses of the Qur'an so how is it possible to rely on the memorization of these early muslims. Sahih Bukhari Volume 6, Book 61, Hadith 558:
Narrated Aisha:
Allah's Messenger heard a man reciting the Qur'an at night, and said, "May Allah bestow His Mercy on him, as he has reminded me of such-and-such Verses of such-and-such Suras, which I was caused to forget."
Further evidence of the Qur'an being altered can be found in other Muslim sources including the tafsir.
Narrated ‘Aasim ibn Bahdalah, from Zirr, who said:
Ubayy ibn Ka‘b said to me: How long is Soorat al-Ahzaab when you read it? Or how many verses do you think it is? I said to him: Seventy-three verses. He said: Only? There was a time when it was as long as Soorat al-Baqarah, and we read in it: “The old man and the old woman, if they commit zina, then stone them both, a punishment from Allah, and Allah is Almighty, Most Wise.” (Musnad Ahmad 21245)
“Aisha narrates: ‘Surah Ahzab contained 200 verses during the lifetime of Prophet but when the Quran was collected we only found the amount that can be found in the present Quran". (Tafsir al Qurtubi)