All perfect praise be to Allaah, The Lord of the Worlds. I testify that there is none worthy of worship except Allaah, and that Muhammad, sallallaahu ‘alayhi wa sallam, is His slave and Messenger.
One should not leave water continuously running when washing the dishes as this is being extravagant and wasteful in its usage. It is confirmed that the Prophet was asked: “Is there any extravagancy and wastefulness when performing ablution?” He replied: “Yes, even if you were (performing ablution) on a running river.” [Ahmad] Moreover, leaving the water running is wasting money. It is confirmed that the Prophet said: “Allaah disliked three things for you: idle talk, wasting wealth, and asking too many questions (about matters that yield no benefit).” [Al-Bukhari and Muslim] For more benefit, please refer to Fatwa 96559.
If a Muslim is going to be accounted for the bounties from which he benefitted, like food and drink and so forth, then how would be the case regarding the bounties he wasted? Allaah Says (what means): {Then on that Day you will be asked about the delight you indulged in, in this world}[Quran 102:8]
A Muslim should not forget the bad state he will face if he was to be deprived of these bounties; one is encouraged to preserve such bounties. Allaah Says (what means): {Say, “Have you considered: if your water was to become sunken [into the earth], then who could bring you flowing water?”}[Quran 67:30]
Therefore, you should close the tap [when unneeded] and you should not be bored by repeatedly doing so. Additionally, if it is possible to put water in a container and then use it for rinsing, then this would be better.