[The avel asked in the shiva house; I gave the in-depth answer afterward.] I understand an avel may do standard learning, such as shnayim mikra v’echad targum (=smvt) on the Shabbat of shiva. Is learning daf yomi also permitted? Answer:
The gemara (Moed Katan 21a) states that it is forbidden to learn Torah during aveilut because Torah study brings true happiness (see Beit Yosef, Yoreh Deah 384). On Shabbat, only some elements of aveilut pertain – the distinction is that devarim sheb’tzina (activities that are done privately) apply (Mo’ed Katan 24a). Rishonim posit that Torah study is a davar sheb’tzina and is forbidden on Shabbat (Rosh, Mo’ed Katan 3:28; Shulchan Aruch, Yoreh Deah 400:1).
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The Rosh and Shulchan Aruch (ibid.) say it is permitted to review the parasha (i.e., smvt). You suggest that this is because any set, regular learning is permitted on Shabbat, i.e., only a somewhat unique study is forbidden. However, these poskim explain their rationale as follows. Since it is obligatory to review the parasha, it is permitted to do so, just as an avel recites Kri’at Shema daily even though it is a form of Torah study (see Nedarim 8a). In other words, it is not seen as learning Torah but just reciting things one normally needs to recite.
Some poskim, starting with the Korban Netanel (on the Rosh ibid.) stress the matter of regularity over the idea that it is not normal Torah study and extend the leniency to any standard daily learning. He says that those with the practice of reviewing 18 perakim of mishnayot daily may do so on the Shabbat of their aveilut. The Aruch Hashulchan (YD 400:6) says that based on this line of logic, those with a set quota of learning, including of gemara, may partake in it on Shabbat. This certainly includes doing daf yomi.
However, this approach is not mainstream. For example, the Pitchei Teshuva (YD 400:3) rules that those with the minhag to learn Massechet Shabbat every Shabbat should not do so during aveilut. The difference is that people are not obligated to take part in this specific Torah activity on Shabbat. (See (Beit Yosef, YD 393 in the name of the Kol Bo; Badei Hashulchan, YD 400, p. 401-3) the parallel discussion of whether an avel should recite Bameh Madlikin on Shabbat. On the one hand, it is a standard part of the siddur; on the other hand, it is Torah for which there is not an obligation, but a weaker minhag, to recite.)
There may sometimes be different grounds to allow taking part in daf yomi on this Shabbat. There is precedent to allow certain type of learning because failure to do so is conspicuous aveilut. The Shulchan Aruch (ibid.) implies that while an avel should listen to Torah reading, he should not receive an aliya even on Shabbat (see Taz ad loc. 1). However, if he was called to the Torah, he must go because his failure to do so on this Shabbat would be considered improper public aveilut. He continues with the story of Rabbeinu Tam, who ruled that since he received shlishi every Shabbat, he should also get it during aveliut for that reason. Arguably, if one regularly takes part in a public shiur in daf yomi or, actually, any other topic, he should not refrain from it on the Shabbat of aveilut to avoid public aveilut.
However, even if one usually takes part in such a shiur, it is hard to consider refraining public aveilut. Consider that the Taz (ibid.) says that an avel who is making a brit on the Shabbat of aveilut should not get an aliya because not always is the father of the brit given an aliya, even though it is standard. It is rare that someone’s attendance at a shiur is more regular than that. If one always gave the shiur, maybe his absence would be starker. Even then, though, one can argue that considering that he has not had much time to prepare all week, his absence would not have to be interpreted as aveilut. (Badei Hashulchan 400:(75) suggests one should appraise each specific occurrence.) In short, it is almost always proper to not learn daf yomi on the Shabbat of aveilut.