Yes, the UN should investigate reports of torture during the attempted coup in Turkey. Any report of torture, regardless of who it may be perpetrated by, should be investigated until its veracity is determined. Governments worldwide must condemn the use of torture, which has been determined to be brutal and largely ineffective.
A UN report found that torture was widespread after the coup attempt in Turkey. It seems to me that under the UN convention against torture, this is certainly something that it should investigate. The report details grim conditions including beatings and rape, detention without food, water or access to lawyers and that may well fall under the definition of torture.
Yes, the United Nation always needs to investigate reports of torture, and that includes the reported cases during the attempted coup in Turkey. As per international law, and as far as basic morality goes, torture is wrong and illegal. Any reports should be taken seriously and investigated to the max.
As one of the United Nations' goals is to protect human rights, they should investigate those reports. They are at full liberty to do so. Who else is capable of policing this issue? This would help to put Turkey in check and those who seek to abuse their power against their citizens.