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 Allaah (peace and blessings of Allaah 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 Allaah (peace
and blessings of Allaah 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 Allaah (peace and
blessings of Allaah 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 Allaah 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 Allaah 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 Allaah 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 Allaah knows best.