Ruling on leaving the Mushaf open when one has finished reading from it

There is nothing wrong with leaving the Mushaf open if you want to come back to read from it, unless that will lead to it being treated in a disrespectful manner or getting dirt or dust or the like on it. In that case it should be closed so as to protect it. However it is better to close the Mushaf when you have finished reading from it, so as to protect it from dust and the like. If there is no reason to leave it open, then closing it is undoubtedly better. 

Al-Hakeem at-Tirmidhi (may Allah have mercy on him) said: 

Another aspect of the sanctity of the Mushaf is that if it is put down, it should not be left open and no other book should be placed on top of it, so that it will always be above other books. End quote from Nawaadir al-Usool, 3/254 

Shaykh Sulaymaan al-Maajid (may Allah preserve him) was asked: 

Is there any report to suggest that it is not allowed to leave the Mushaf open when one goes and leaves it? 

He replied: 

We do not know of anything in Islam to suggest that it is not allowed to leave the Mushaf open; similarly we do not know of anything in custom to suggest that this is disrespectful towards it. Based on that, there is nothing wrong with it, although it is preferable, in order to be more careful with it, to close it so as to protect it from any liquid or dust. And Allah knows best. End quote. 

And Allah knows best.