I am aware of the fact that a Sunni marriage will not be considered valid if there are no witnesses for the marriage. As far as I know, there are ahadith which stress on the necessity of this condition.
I haven't found verses in the Qur'an yet about witnesses in marriages but I found ahadith on IslamQA with regards to this matter:
In order for a marriage to be valid, it is stipulated that it be witnessed by two Muslim witnesses of good character, because the Prophet (peace and blessings of Allaah be upon him) said: “There is no marriage except with a wali (guardian) and two witnesses of good character. Narrated by al-Bayhaqi from the hadeeth of ‘Imraan and ‘Aa’ishah; classed as saheeh by al-Albaani in Saheeh al-Jaami’ no. 7557.
Shi'a Muslims, on the other hand, claim that it is mustababb for the witnesses to be present but not wajib. I believe that any Islamic ruling should be firmly backed by the Qur'an and the sayings of Prophet Muhammad (PBUH).
Relevant statement from the link above:
The Imamiyyah do not consider the presence of witnesses as wajib but only mustababb.
The following is stated regarding marriage witnesses on Al-Sayyid Ali Al-Husseini Al-Sistani's website:
It is permissible for the couple to recite the formula of marriage agreement by themselves or by appointing representatives who will recite it on their behalf. There is no condition for the presence of witnesses during the solemnization of the marriage, just as the presence of a cleric is not a condition for the validity of the marriage.
On a website called Alulbayt Foundation, the following answer (and question) is given for marriages:
Q14) Can a permanent marriage be made secretly without the permission of parents of bride and without the witness being called?
A14) Presence of two witnesses is not necessary when a marriage contract is being read. And a girl who is not independent in her living affairs should obtain permission of her father or paternal grandfather for marriage. And if she is independent, she should obtain her guardian’s permission as a measure of obligatory precaution.
On Wikipedia, for the topic of Islamic marriage contract, the following has been stated about witnesses for Shi'a marriages:
In Shia Islam, witnesses to a marriage are deemed necessary, but in case are not available then the two parties may conduct the nikah between themselves.
Which Qur'anic verses and/or ahadith support the Shi'a ruling on the presence/absence of witnesses for marriages?