If a person touches an impurity that is dry, it is not transferred to him and neither his clothing nor his body become impure as a result. Impurity is only transferred when there is wetness.
Shaykh Ibn Jibreen (may Allah preserve him) said:
If dry impurity touches the dry body or clothing, it does not matter. The same applies to entering the dry bathroom barefoot, if the feet are dry, because impurity is only transferred when it is wet. End quote.
Fataawa Islamiyyah, 1/233
Shaykh Saalih al-Fawzaan (may Allah preserve him) said:
If a person touches impurity that is damp, then he should wash from his body whatever he has touched, because the impurity has been transferred to it. But if the impurity is dry, he does not have to wash off what he has touched, because it is not transferred to him. End quote.
Al-Muntaqa min Fataawa al- Fawzaan, 48/18
It should be noted that the mere presence of dampness does not mean that the impurity is inevitably transferred. Rather there may be a little dampness which does not mean that the impurity is transferred. Hence the Hanbali fuqaha’ stipulated that in order for damp impurity to be transferred, there should be actual wetness. If there is no actual wetness, then the impurity is not transferred even if there is dampness.
See: Kashshaaf al-Qinaa’, 1/185
What appears to us to be the case is that this impurity that you have asked about is very little, and we doubt that it was transferred to the furnishings, because the dampness that accompanied it was very little. Even if we assume that it was transferred to the furnishings, it would soon have disappeared and there will be no trace left of it.
Shaykh Ibn Jibreen (may Allah preserve him) was asked about a man who always advised his wife and his sons and daughters and dependents that when they did wudoo' in the washroom they should not walk on the carpets with wet feet because lest there be some urine in some places in the house the exact location of which is not known, to prevent the urine attaching itself to their feet and making the prayer invalid, but they did not pay any attention to that because it is too difficult and they did not think that this was an important matter, because the house was big and there were a lot of furnishings. What is the solution? May Allah reward you with good.
If the impurity that is on the carpets is not clearly visible, then it seems that it does not affect anyone who steps on it.
Moreover, what usually happens is that when a person washes his feet after wudoo' and then steps on the carpet, the carpet quickly dries whatever wetness is on his feet at first, so it is possible to wash the mats that are placed close to the bathrooms to dry the feet after wudoo', and after that it will not matter if one steps on the contaminated carpets because the feet are no longer damp after stepping on the carpets that are nearest to the bathrooms. There is no need for this strictness in telling people not to step on those impure carpets. You can put towels at the doors of the bathrooms which will absorb the dampness from the feet after finishing wudoo', and after that any slight dampness on the feet will not matter and will not invalidate the prayer or transfer impurity. End quote.
Thus it is clear that there is no need for this strictness or to take the matter to this level of harshness and pressure. These furnishings cannot be deemed impure so long as the impurity cannot be clearly seen on them and has left no traces.
And Allah knows best.