I've noticed that Reform and Conservative often use small "scarf" talleisim (here's an image). Where did this practice originate? (Is there any connection to the scarves that some Christian priests wear, and if so, could there be an issue of chukas hagoyim)? Is there some ideological reason why Reform and Conservative continue using "scarf" talleisim b'davka, or is it just something that they are used to doing for historical reasons?
According to this link, which I received in an answer to a question about Spanish-Portuguese minhagim, the "scarf-like" tallit (or tallet) is originally from their community. As you can see below, the ba'alei battim, the chazzan, and the chacham all wear the style in question.
As the other answers indicate, the scarf way to wear it happened in certain other communities as other times. I haven't seen any point of connection between the modern practice and these isolated instances. The most likely origin of the Ashkenazi Reform/Conservative way of wearing it as a scarf is the imitation of the Christian Stole. The Reform synagogue, in an effort to assimilate, copied many of the features of Protestant Church. Rabbis would, and still do in many synagogues, wear Clerical Robes and fold their tallith to resemble a stole so they could closer resemble a Protestant Minister. Add in a choir, organ, and vernacular services, and superficially, an early reform service would have been hard to differentiate from a German Protestant church service. My best guess for why it became widespread is that congregants would see their rabbis wearing the tallith in such a manner, and copy it.
i can't speak to its exact origins, but it seems to have been a well established practice by many Jewish communities scattered all over the world. This is the official Halacha of Egyptian Jews regarding Tallitoth.
Nehar Misrayim Chapter 1:
במצרים רובא דרובא דהמוני עם משלשלין כל הארבעה ציציות לצד פניהם והטלית מקופל על צוארם וראוי להודיעם שלפחות בשערת העטיפה. יברך ויתעטף בו ראשו ורובו ויעמוד כן כדי הילוך ארבע אמות ואח״כ ישלשלנו על כתפו. אבל אם מתחילת הברכה מנחו מקופל על כתפו הוי ברכה לבטלה דהרי ליכא עיטוף. וגם מצות ציצית לא קיים. והמוני עם מסתפקים בנשיקת הציצית פעמים ושלש ובזה חושבים לפיים את הציצית על אשר לא טוב עשו לו. וראוי להוכיחם ולהשיב מטעותם שלא יכשלו באיסור ברכה לבטלה. וד״ב.
In Egypt the most common way is that one drapes all four ṣiṣioth to their front and the tallit is folded on their neck. And it's appropriate to inform them in their time of wrapping. He blesses and wraps his head and most of his body in it and will stand up in the space of four cubits and afterward will drape it over his shoulder. But if from the beginning of the blessing he lays it folded on his shoulder, then it is for nothing. And the commandment of ṣithṣith has not been carried out. And the common people suffice in kissing the ṣithṣith twice or three times and with that they think they make up to the ṣithṣith for having mistreating it. They should be reproached and their mistake should be clarified to prevent them from failing again in the prohibition of saying a blessing in vain.
Here is a photo that was brought back from my family from Cairo.
from what I have heard the original reform movement made change in in davening both the liturgy and the dress of those leading to be more like the church. This was part of their goal to become assimilated and then so they thought become more accepted by non-Jews. Today I don't think this people today use this tallis for this reason but rather after several generations it simply became popular. The main problems with such tallesim... 1)they often aren't big enough to require tzitzis. 2)even if they are large enough wearing them in such a manner is problematic as each of the tzitzis are to be on the four corners around one's body and not all in the front.