There are several issues in the 1 Cor 7 passage, it would be helpful to know exactly which verses are being referenced.
First of all, if someone is making a claim about a law being of no effect, or being repealed, the burden of proof is on the individual making the claim. If you can post a summary of the person's claim, we can better understand their thoughts.
Secondly, God's Laws cannot be repealed, they can only be fulfilled, and only by God Himself.
Thirdly, those two passages do not conflict, so by default are not in opposition to each other, or capable of repealing each other, or over-ruling each other.
The Corinthian passage talks about avoiding fornication by being committed INTO marriage, and that if a believing woman leaves, she is still bound to her husband, and the husband must take her back if she returns.
1Co 7:2 Nevertheless, to avoid fornication, let every man have his own wife, and let every woman have her own husband.
1Co 7:11 But and if she depart, let her remain unmarried, or be reconciled to her husband: and let not the husband put away his wife.
The Matthew 19 passage in similar fashion says that unless a woman commits fornication, the husband is not allowed to put her away.
Mat 19:9 And I say unto you, Whosoever shall put away his wife, except it be for fornication, and shall marry another, committeth adultery: and whoso marrieth her which is put away doth commit adultery.
In both passages, the two are still married and are not allowed to separate, and there is nothing to repeal because the two passages are in agreement. In the Corinthian passage, it is clearly stated that she is to NOT marry, but must return to her husband. She cannot because she would have to commit fornication/adultery in order to marry again, because no matter where she is geographically, she is still a married woman.
This verse is referencing a believer that is married to a spouse that is NOT a believer and leaves. A woman that believes is left by a man that is not only gone, but not willing to be a husband or a covering for her. She is not bound to him any longer, she is free to marry again.
1Co 7:15 But if the unbelieving depart, let him depart. A brother or a sister is not under bondage in such cases: but God hath called us to peace.
The Matthew 19 passage has nothing to do with believers married to unbelievers, and therefore cannot be applied directly against 1 Cor 7:15. The passage in Matthew is about "divorce for any cause", which was the practice of the jews of that time.