It seems useful to compare such acts to masturbation. Masturbation is generally disapproved of within Christianity, but there is not universal agreement on the rationale for this prohibition. This opacity has a lot to do with the Bible saying almost nothing about masturbation. Wikipedia mentions the story of Onan in Genesis 38, but the relevance of this passage (in which Onan is pressed by his father to sleep with his brother's widow, but refuses to impregnate her) seems dubious.
Leviticus 15 discusses the way a person may be made ceremonially unclean due to discharges of semen. However, I would presume that this no more applies to Christians than the Old Testament regulations concerning women's periods. Jesus' death removed the need for this kind of ceremonial purity, substituting his purity in the place of men.
So much for attempting to get an answer directly out of scripture. I am aware of four arguments as to why masturbation is usually or always sinful, which we shall refer to as the Purity, Radically Pro-Life, Teleological and Lust arguments. Note that accepting one does not entail the rejection of the others, and masturbation might be sinful for multiple reasons. There are, though, good reasons for rejecting at least the first two of these arguments.
This is broadly the argument I suggested referring to Leviticus. The claim would be that masturbation in some way defiles or profanes the body. The Christian's body being a temple to the Holy Spirit and all, masturbation is therefore wrong.
Why, however, would we think that masturbation is profane in this way? As Paul writes in Romans 15:14:
I am convinced, being fully persuaded in the Lord Jesus, that nothing is unclean in itself. But if anyone regards something as unclean, then for that person it is unclean.Masturbation might on this view be wrong for some people. But there's nothing here to justify a blanket prohibition.
Radical Pro-Life Argument
Life, many Christians claim, begins at conception. (I think a better way to put this would be that "the distinctively valuable feature of life - i.e. being part of God's plan - begins at conception", but that's a debate for another time). Why not go further, then? Why not argue that not merely a fetus, but even a sperm, has moral importance and should be treated with the same reverence as a person?
The first thing to note is that this would only establish a prohibition on masturbation by men. (And indeed, men would be perfectly entitled to masturbate so long as they did not ejaculate), Female masturbation is hardly going to kill an egg.
But even with that restriction, the argument seems dubious. An embryo will, in the right conditions, grow into a person: it is, one might well believe, a person who has yet to fully develop. A sperm is not like this, however, since it needs not only the right nourishment and protection to become a person, but also to be united with an egg.
Moreover, even when an act of sex does lead to procreation, there are hundreds of millions of sperm which do not fertilize the egg but instead die within a few days. If God had imbued sperm with intrinsic moral value, would He really allow 99.999999% of them to die even in the best case?
This was for a long time - perhaps still is, I don't know - the official view of the Catholic Church. According to this view, the purpose of sex is procreation within marriage. All acts of sex which do not aim towards this purpose are sinful.
I'm not going to challenge this as an argument, given that it has been developed and defended over hundreds of years by minds greater than my own. Perhaps it succeeds, perhaps it does not, and it would be the height of arrogance to think that I can refute its strongest form without ever having looked into it before now.
Two things are worth noting, however. First: if you accept this argument you should also oppose a wide range of other sexual acts, including not only familiar sins such as sex outside of marriage and homosexual sex but also sex using contraception, ejaculation into any orifice other than the vagina, sex between couples in which the woman is post-menopausal or already pregnant, etc.
Second, one can reject this argument while maintaining that sex has a particular purpose of procreation within marriage. It is mainstream doctrine in many denominations that sex is intended not only for procreation but also for pleasure, just as food is intended both for our pleasure and for our sustenance. Moreover, unless one thinks that whatever serves God's purposes is mandatory and whatever does not serve His purposes is forbidden, there remains work to be done in the move from "sex is intended by God for procreation by married couples" to "all sexual acts which cannot lead to procreation are forbidden."
This, I think, is by far the most plausible argument against masturbation. Consider Matthew 5:27-28:
Masturbation is not in itself sinful. However, it is usually predicated upon, or at the very least assisted by, a sinful lust. Masturbating to thoughts of someone to whom you are not married is a violation of the Seventh Commandment. If one is able to masturbate without thinking of anyone, then sure, go ahead; if one masturbates thinking of one's spouse, then again go ahead (although procreative sex is perhaps better!).
What does this mean for sex with robots?
The Purity argument would presumably apply to robots as well as masturbation. But as we saw, this will make you impure only if you think it does and decide to go ahead anyway.
The Radical Pro-Life argument would establish a prohibition upon men ejaculating as a result of sexual relations with robots. We might think that for men to engage in sex with robots at all is therefore spiritually unhelpful and conducive to sinning, even if not sinful in itself. But as noted, this argument seems highly dubious and could not in any case establish an objection to women making use of sex robots.
The Teleological argument would presumably establish a prohibition on the use of sex robots, except where such usage is conducive to procreation. Again, though, the argument seems dubious.
What of the Lust argument? Does lust towards a robot constitute adultery? If the robot is intended to represent a particular person to whom one is not attracted, then it seems that it should: one would not be excused lusting over a pornographic actress merely because technically one lusted over a picture of her on a screen, so it should hardly be different if we replace the screen with a 3D representation.
Following this, I think it makes sense to think that any sexual act with a robot in which the act is reliant upon the robot's being representational of a person to whom one is not married should be considered sinful. Sex with a stranger is adultery just as much as sex with a known person. That said, a robot which is representational of one's spouse does not seem to fall foul of this rule: if sexting and sending of naked selfies between married couples is permissible, then masturbation to 3D representations of each other ought also to be permissible. Additionally, robots which produce sexual pleasure without being representational of any person would not fall foul of the lust argument.
I'm not here to set your doctrine for you. Perhaps you think I am wrong to reject the Teleological argument, in which case sex robots can be almost completely ruled out. Even the Lust argument that I think succeeds sets some strong limits on what kinds of sex robots can be used without sin. But it seems mistaken to argue that Christians should accept a blanket prohibition upon all such uses of robots.