Is it permissible for a non-Muslim to enter Madeenah?

Firstly: 

It is not permissible allow the kuffaar to take up residence in the Arabian Peninsula. The scholars differed at to the definition of the boundaries of the Arabian Peninsula, but they did not differ as to the fact that Madeenah is part of it. 

Ibn Qudaamah said: 

It is not permissible for any of them (the kuffaar) to live in the Hijaaz. This is the view of Maalik and al-Shaafa’i, but Maalik said: I think that they should be expelled from all the Arab lands, because the Messenger of Allah (peace and blessings of Allah be upon him) said: “Two religions cannot coexist in the Arabian Peninsula.” Abu Dawood narrated with his isnaad from ‘Umar that he heard the Messenger of Allah (peace and blessings of Allah be upon him) say: “I will certainly expel the Jews and the Christians from the Arabian Peninsula,, and I will not leave anyone there but Muslims.” Al-Tirmidhi said: this is a saheeh hasan hadeeth. And it was narrated that Ibn ‘Abbaas said: The Messenger of Allah (peace and blessings of Allah be upon him) left behind three instructions. He said: Expel the mushrikeen from the Arabian Peninsula, honour the delegations the way that I do, and he kept quiet about the third. Narrated by Abu Dawood. 

Al-Mughni, 9/285, 286. 

Secondly: 

It is permissible for kuffaar to enter Madeenah for the purposes of trade, without staying there. They should be given enough time (to complete their business) then they should be told to leave. Ibn Qudaamah (may Allah have mercy on him) said: 

It is permissible for them to enter the Hijaaz for the purpose of trade, because the Christians used to trade with Madeenah at the time of ‘Umar (may Allah be pleased with him). An old Christian man came to him in Madeenah and said: “I am an old Christian man and your agent has taken the tithe from me twice.” ‘Umar said: “I am an old monotheist (haneef) man,” and ‘Umar wrote to the agent and said: “Only take the tithe once in a year, and do not allow them to stay for more than three days.” This is what was narrated from ‘Umar (may Allah be pleased with him) and they should leave after that. Al-Qaadi said: Four days is the limit after which a traveller must offer prayers in full. 

Al-Mughni, 9/286. 

Thirdly: 

What we have mentioned about Madeenah and its sanctuary does not apply to the sanctuary of Makkah. The kuffaar are forbidden to enter it under any circumstances. 

It says in al-Mawsoo’ah al-Fiqhiyyah (3/130, 131): 

The majority of scholars – including Muhammad ibn al-Hasan among the Hanafis – are of the view that it is not permissible for kuffaar to enter the sanctuary of Makkah at all, The view of the Hanafis is that that is permissible if there is a treaty or they have permission. 

With regard to the sanctuary of Madeenah, it is not forbidden for them to enter to bring a message, for trade or to bring some goods. A kaafir may not enter other parts of the Arab lands without permission or a treaty. The fuqaha’ have discussed this matter in detail.Ends.

 And Allah knows best.