What is the ruling of Islamic law on those who claim that the Prophet Muhammad (peace be upon him) visits their living saints these days?

The best way to respond to innovations (bid'ah) and correct errors is to request legitimate evidence, as religion is based on sound proofs from the Qur'an, Sunnah, and scholarly consensus (Ijma'), not on anecdotes and reckless claims.

Whoever claims to have seen the Prophet (peace be upon him) while awake must provide proof, either by claiming that he is alive in both spirit and body, or that his image is revealed to them in his intermediate life (barzakh). Scholars have investigated these claims and found nothing but stories and anecdotes from individuals, which do not constitute legitimate evidence.

Moreover, these stories are subject to many possibilities such as lack of verification, delusion, occurring in a dream and not while awake, or the devil appearing in the form of the Prophet (peace be upon him).

The Companions and those who followed them have unanimously agreed that the Prophet (peace be upon him) died and ascended to the Highest Companion. This is supported by the statement of Abu Bakr Al-Siddiq (may Allah be pleased with him): "Whoever used to worship Muhammad, then Muhammad has died."

Shaykh al-Islam Ibn Taymiyyah indicated that what happens to some people regarding seeing prophets and righteous individuals while awake is from the actions and whispers of devils.

Scholars also mentioned that the Companions, who were the most beloved people to the Prophet (peace be upon him), did not see him after his death while awake, nor did they ask him about matters they differed on after his death. Had it been possible, they would have done so.