There are many Muslims who claim that there are many pagan influences which shaped the Christian faith and corrupted it. Here some answers how serious these speculations should be taken.
Most of these pagan origin theories were very popular around 1900 but have been discarded by NT scholarship long ago. Only some atheist groups and Muslims seem to be stuck on it but none of the experts of early Christianity find this acceptable anymore (including those who are not Christians themselves).
"Christmas trees are pagan idolatery" and answers to that rather regular charge: [*, *, *]
There exist quite a number of works on the pagan origins of Islam. Some of them can be found on the web. We are no experts on this issue, and cannot judge the scholarly merit of the following material, but it might at least help to see the issue in a new perspective given that similar arguments are made for Islam. Therefore Muslims need to be more careful when they want to use these ideas as it can easily be turned on Islam in the same way.
Ronald Nash, The Gospel and the Greeks (Dallas: Word Publishing 1992)
Arland Hultgren, The Rise of Normative Christianity (Fortress Press 1994)
Leon McKenzie, Pagan Resurrection Myths and the Resurrection of Jesus
Ancient paganism and The Paganism that is Modern Day Christianity are two representatives of the Muslim sites claiming that some superficial similarities are the same as "origins" even when the Christian doctrine is older than these pagen beliefs. But in general, the Muslims might think a bit harder on the question whether similar beliefs before Christianity or Islam respectively mean that it is borrowed and therefore false.
Even Muslims found themselves the necessity to write on Religious Borrowing since a large part of the Qur'an is very closely or nearly verbatim taken from various non-Islamic sources. You may consult in particular Tisdall's book "The original Sources of the Qur'an" on this topic. More about this topic in the section on the Sources of the Qur'an.
Do we have again two standards of measuring? One for Christianity and one for Islam?
Most material on the following pages is pretty laughable, but in order to raise awareness of some who actually believe this:
Zoroastrian Influences on Judaism and Christianity and Monotheism
The Origins of Christianity and the Quest for the Historical Jesus
Muslims, don't rejoice to much in this: If Jesus never existed as claimed for example in the second of the above articles, then the Qur'an is as wrong as the Bible. The same holds for the following which some Muslims have jumped to a bit prematurely:
Allegedly, the Roman Piso Family invented Jesus and all of Christianity (1, 2). Responses from a Christian newsgroup. Here is a much more detailed response to the Pisomania by J.P. Holding. Interestingly, the same group also included the Qur'an into their theories (here).