Office of Grand Ayatollah Khamenei (may Allah grant him long life):

There is no problem in it. Generally speaking, there is no objection in opening and closing tap before washing the left hand completely.  However, if one touches a wet thing (such as a wet tap) and external water gets mixed with Wudhu water after completing Wudhu and washing the left hand, it is not in order to wipe the head and feet with external or mixed water.

Office of Grand Ayatollah Makarem Shirazi (may Allah grant him long life):

There is no problem in it.

Office of Grand Ayatollah Saafi Gulpaigani (may Allah grant him long life):

Preventing prodigality and economic use of water in the said manner is desirable and there is also no problem if one closes and opens the tap several times. Yes, after washing the left hand and completing Wudhu, do not touch external water with your fingers; otherwise wiping the head and feet with the finger which has come in contact with external water is not correct.

Ayatollah Mahdi Hadavi Tehrani’s answer to the above question is as under:  

1. It is haram to be prodigal.

2. It is mustahab (recommended) to pour much water on the Wudhu parts which are washed during ablution.

3. It is permissible to close tap amid the Wudhu acts; rather it is obligatory to close it in order to avoid prodigality.

4. While performing Wudhu and after washing the left hand, one’s hand must not come in contact with external water so that he may not have to wipe his head or feet with external moisture. Therefore, if the tap is wet and its moisture is more than the moisture on the fingers, it is necessary that the part of the hand which is used to wipe the head and feet should not have come in contact with the tap so that wiping is done with the moisture left from Wudhu water.

Link to Isteftaat Website (Code: 972)