You're not allowed to take holy items into the bathroom/wash-house. If a newspaper has mostly advertisements but it also has a few pages of divrey torah, are you allowed to take such a newspaper into the bathroom granted that you will not be reading the divrey torah?
There is a dispute whether modern-day toilets are like the bathroom of the gemara. Quite a number of poskim are strict.
A newspaper that contains divrei Torah should not be brought into the bathroom (see Igros Moshe Y.D. 2:76, Be’er Moshe 3:183, 8:127, Oz Nedberu 2:64, Teshuvos V’hanhugos 2:466, Ginzei Hakodesh 8:footnote 14, see 8:9)
Some say that it can be brought in as long as the page with Hashem’s name is not opened in the bathroom, since the pages of the paper are considered a covering (see Tzitz Eliezer 11:5, Ginzei Hakodesh 8:footnote 16 quoting the opinion of R Elyashiv and others)
Others (R Jonathan Blass and R David Sperling) write that one only need to worry if God's name is written in Hebrew and if one saw it in the paper. It is in any case forbidden to go to the bathroom with the intent to read/learn words of Torah. See references above and here for further sources.
Some people are strict and only keep secular books in the toilet, such was the practice of R Eliyahu Moshe Shisgal (the son in law of R Moshe Feinstein, see artscroll's biography of R Moshe). There is also a rumor (probably false) that the Vilna Gaon wrote his sefer on geometry (Ayil Meshulash) on the toilet.