Answer: The Gemara (Shabbos 5b) writes that if a sefer unrolled into a reshus hayachid in a way that they couldn’t roll it back, they should turn it over so that the writing should not be exposed.
The Shulchan Aruch (YD 277:1) writes that when a sofer who’s writing a sefer Torah leaves it to dry, he should cover it rather than turn it over, as covering it is more respectful. The Bach (YD 277:1) writes that we learn from here that it is disrespectful to leave a sefer Torah or other sefarim open for any extended period of time (See Shach YD 277:1; Baer Heitev YD 277:1; Aruch Hashulchan YD 277:2).
Thus, the Shulchan Aruch (OC 139:5) writes that the minhag is to cover the sefer Torah between each aliya. The Rema, however, writes that the minhag ashkenaz is to roll it closed. Likewise, the Mishna Berura (139:21) quotes the Taz that one shouldn’t cover the sefer Torah each time as it is a tircha, extra bother.
The Baer Heitev (OC 139:8) adds, however, that it is appropriate to cover the sefer Torah when saying kaddish afterwards. Thus, we cover the sefer Torah whenever there is a substantial pause, such as the gabbai saying mi shebeirach.
In conclusion, one should close, but not cover the sefer Torah after an aliya. When there is a longer break, such as kaddish or mi shebeirach being recited, then it should be covered.