Palestinian Christians specifically do face attacks from extremist settlers (who don't care that they're not Muslim; they care that they're not Jews). However they also face oppression from islamists. there's a reason for substantial Christian representation in secular Palestinian nationalist movements.
christians are also more likely to emigrate. (By the way this is also true in Lebanon, so the phenomenon is more complicated than Israeli-Palestinian politics.) Bethlehem, for example, has seen a huge change in demographics; it's now 90% Muslim where it used to be majority Christian. This is not due to settler violence but the economic and social impact of the occupation and its particular geographic position (it's cut off between settlements and the separation barrier, and before that by checkpoints.) The settlers in this area are not violent, in fact they pride themselves on not being (though they are willfully blind to the impact of their presence).
The situation in Jerusalem itself is complicated and always has been, and there are layers beyond the conflict itself such as disputes between different Christian denominations. When it comes to Christians in general, not specifically Palestinians, there's also been several cases of Christians trying to infiltrate the Jewish community, that is, missionaries posing as Jews. This is an incredibly sensitive issue in Israel. (It is not illegal to be a missionary, but it is such a sensitive issue that when BYU, the Mormon university, opened a Jerusalem center, the students were given a specific order not to evangelise and it is very strictly enforced.) When one of these incidents happens, it can result in reactions towards the Christian community.