What is the Biblical basis for it being wrong for couples to live together “chastely” before marriage? Many Christians say that a couple shouldn't live together before they get married, even if they are not having sex with each other.

What is the Biblical basis for this position? Are there certain passages of Scripture that speak against it? Or are there aspects of Church tradition that speaks against it? Or is there something else that has prompted Christians to view this as wrong?
From your comments you are asking strictly about members of the opposite sex who live in the same house, without any kind of sexual relations.

There is, as far as I know, no absolute prohibition on this. In fact I have known several Christians who have shared houses with members of the opposite sex and nobody has objected in any way. However I do understand your claim that Christians do sometimes object to this. Here are some reasons:


They don't believe it. The church doing the objecting probably thinks the pair are having sex, they are just not being open about it. If the people sharing the house are in a romantic relationship, this skepticism is pretty understandable. 
It's laying yourself open to temptation Having someone of the opposite sex 'available' and in a situation where nobody would know if you slept together is heightening the possibility that you might 'slip up'. If you are attracted to the other person that counts double, and if you are in a romantic relationship with them, even more so. 
It's damaging your reputation and hence Jesus'. Even if you are living completely chastely, and fellow-Christians believe that, it's extremely unlikely that the rest of the world will believe it. Therefore they will see it as a case of Christians being hypocritical.


Different churches would have different takes on this. I certainly know some that would have have no problem with it, and some that would have a big problem.