TheLady and
narzos have quite thoroughly established that
Jerusalem is definitely
considered holy in
Islam, but
it might still be worth pointing out
why this is so.
First and foremost, while the Qur'an is the supreme Islamic holy book,
it is not the only one: the Torah and the New Testament are
also holy, and so are the prophets described therein. Thus, if
Jerusalem is holy to a Jew or a Christian, it is also holy to
a Muslim. (This is a bit of an oversimplification since the
Qur'an overrides previous texts on a few key issues like Christ's
divinity, but this is the "default" setting, so to speak.)
Now, you will recall that Abraham offered to sacrifice Isaac
(or Ishmael)
on Mt. Moriah, and by tradition Solomon's Temple -- and hence
the present day Haram al-Sharif -- was built on this spot.
This story is even explicitly retold in
Sura 37:102 of the Qur'an.
The location of the sacrifice is not mentioned, and some Islamic
scholars think it was in Saudi Arabia, but by most traditions
the spot was in Jerusalem. (And for what it's worth, neither the
Torah nor the Bible specify the location well enough to make a
call on this.)
Second, while the Qur'an does not mention Jerusalem by that name
(which is not particularly surprising since Yerushalayim is
Hebrew), Sura 17 -- which also contains the famous verse
cited above -- says quite a bit more about a familiar-sounding
place. The land was given to the Children of Israel by Moses (17:2),
the Scripture warns that it will be destroyed twice (17:4),
and so it was (17:5, 17:7). It being fairly well established that
the Promised Land is Israel, it doesn't take too great a leap
of faith to connect these two "destructions" to the razings of
Jerusalem by the Babylonians and the Romans.
Additionally, in the Hadith Muhammed often refers to Bayt al-Maqdis
("The Purest House") as the destination of his Night Journey:
Abu Hurairah, may Allah be pleased with him, reported:
Allah's Messenger (may peace be upon him) said:
I found myself in Hijr and the Quraish were asking me about my Night Journey. I was asked about things pertaining to Bayt Al-Maqdis which I could not preserve (in my mind). I was very much vexed, so vexed as I had never been before. Then Allah raised it (Bayt Al-Maqdis) before my eyes. I looked towards it, and I gave them the information about whatever they questioned me.
Again, some scholars think that the Bayt is only an abstraction of a
perfect city, not a physical place. What is certain is that the label
is often used for Jerusalem, even among Arabic-speaking
Christians and Jews, and has been used as such by people like
Saladin and Omar ibn al-Khattab, two of its many conquerors.
Third, since the Holy Mosque of 17:1 is unanimously agreed to be the mosque
of Mecca, it would seem rather odd to label Medina the
"farthest" place of worship -- the two are only some 250
miles apart from each other, not all that great a distance on
the Arabian scale of things as a camel can travel that far in
less than two weeks. Remember, Muhammad himself had traveled
as far as Syria, well over one thousand miles away.
Now, none of this constitutes ironclad "proof" of Jerusalem's
holiness, but as TheLady wisely pointed out Judaism and
Christianity can't "prove" holiness either. But the reasons outlined
above have been meaningful enough to make Jerusalem the third most
holy city in Islam, and dissenters from any are a minority view.
References
The Holy Qur'an
http://members.aol.com/bazbooz/BaytAlMaqdis.html