According to the Shia view of Islam, is mut'ah marriage allowed with one's daughter, one's sister or one's sister-in-law?
All of these are obviously forbidden, because the Qur'an says:
Prohibited to you [for marriage] are your mothers, your daughters, your sisters ... And [also prohibited are] ... you take [in marriage] two sisters simultaneously ...
Matrimonial relation is forbidden with women who are considered one’s Mahram, for instance, mother, sister, and mother-in-law.
And who a man may marry is the same in indefinite and fixed-term marriages:
There is no difference between the indefinite and the fixed-term marriage except [mahr, inheritance, living expenses, equal share in the case of polygamy, and waiting period after divorce] Islamic Law, Ayatollah Shirazi (pdf)
So it's going to be what you would naturally expect. Nevertheless, here's Shia scholars enumerating these relatives directly:
It is forbidden to marry some of the women such as the mother, sister, and mother-in-law.
Normally, non-mahram male and female can marry; the exception is – for example – while married to his wife a man may not marry his sister-in-law (his wife’s sister).
Islamic Law, Ayatollah Shirazi
Matrimonial relation is haraam with women who are one's Mahram, for instance, mother, sister, daughter, paternal aunt, maternal aunt, niece (one's brother's or sister's daughter) and mother-in-law.
Women with whom matrimony is Haraam, Ayatollah Sistani
... except for certain categories of woman whom he cannot marry for they are forbidden due to blood relationship. These are: ...
- Sister-in-law, at the lifetime of his wife, for it is forbidden to marry two sisters at the same time.
Jurisprudence Made Easy, Ayatollah Sistani
In the name of Allah
Briefly speaking, Nikah al-Mut'ah has similarity with permanent marriage in many laws -- although it has its specific conditions, too. Meanwhile, it needs much explanation to elucidate the issue completely, but, in short, (AFAIK) it is haram to married to:
Maharems, such as:
Mother, sister, mum-in-law, sister-in-law, aunts and so on.