They have someone with them who is incontinent, and he sits on the rugs; what is the ruling on praying on them?

If the sick person is certain that the urine has ceased, there is nothing wrong with him taking off the diaper and sitting on the rug. 

The basic principle is that it should be assumed that he is telling the truth, but if you find out that it is not as he says, and that some of his urine is getting on the rugs, then you should be cautious with regard to prayer, so each person should wash his feet before praying lest any impurity has gotten onto them, and he should use a clean rug to pray on. 

The basic principle with regard to these rugs is that they are taahir (pure), and they cannot be ruled to be impure just because there is some doubt. But if it is proven that there is some impurity and it is known where it is, and it is to difficult to pick it up and clean it, then it is sufficient to pour some water on the spot where the impurity is in order to clean it, pouring enough to cover it. 

The Standing Committee for Issuing Fatwas was asked: 

If a child urinates on the rug, depending on his age, is it sufficient to pour water on it to cleanse it of impurity? Because the rug may be large or it may be a fitted carpet, or there may be large pieces of furniture on it etc.  

They replied: 

If the urine that is on this rug etc is from a boy who is not yet eating solid food, then it is sufficient to sprinkle water on it in order to cleanse it, until the area where the impurity is has been covered, and it is not essential to wring it out or wash it. If the boy is eating solid food or it comes from a girl, whether she is eating food yet or not, then it must be cleansed by washing, and it is sufficient to pour water on the place where the impurity is, and it is not essential to remove the rug or wring it out; this is like impurity on the ground, because it is proven in al-Saheehayn from Anas ibn Maalik that a Bedouin urinated in the mosque and the Prophet (peace and blessings of Allaah be upon him) ordered that a bucket of water be poured over his urine. End quote. 

Fataawa al-Lajnah al-Daa’imah (5/364). 

And Allaah knows best.