Avel/Mourner receiving an Aliyah during Shiva:
During the weekdays: It is forbidden for an Avel to receive an Aliyah throughout Shiva. This applies even if the Avel is a Kohen and there is no other Kohen available in the Minyan. [Rather a Yisrael is to be called up in his place.]
On Shabbos: It is forbidden for an Avel to receive an Aliyah even on Shabbos. If, however, he is the only Kohen [or Levi] available in the Minyan, then he may receive an Aliyah. [Some Poskim, however, rule that it best for the Kohen to leave the Shul and have a Levi or Yisrael called up in his place if this is not uncommon practice in the Minyan. It is however disputed amongst Poskim if this likewise applies to a Levi. If the Kohen or Levi does not leave the Shul then he is to be called up.] Likewise, in any case in which not getting an Aliyah on Shabbos will cause the mourning to be publicized to the Minyan, it is an obligation for him to get an Aliyah. Thus, if the Avel was accidently called up for an Aliyah on Shabbos, he must go up. Likewise, if he is accustomed to always receive a certain Aliyah, then he must go up for that Aliyah even if the Gabbai does not call him by name. [Based on this, the Rebbe was accustomed to receiving an Aliyah on Shabbos during Shiva, as he did every Shabbos.]
Q&A on Aliyah during the week
During the week, if the Avel is a Kohen/Levi is he to leave the Shul prior to the Aliyah?
If the Avel was accidently called up for an Aliyah during the week, is he to go up?
No. This applies even if he is a Kohen. However, some Poskim rule he is to go up for the Aliyah.
Q&A on Aliyah during Shabbos
May an Avel during Shiva receive an Aliyah if he has a Chiyuv, such as a Yahrzeit, or a Baal Bris, and the like?
May an Avel receive an Aliyah on Shabbos day if it is the 7th day of Aveilus?
Shacharis: Although from the letter of the law he is allowed to receive an Aliyah during Shacharis, practically, he is not to do so.
Mincha: He may receive an Aliyah by Mincha.
May an Avel prior to Shiva receive an Aliyah during Simchas Torah?
If the Minyan is made up of Aveilim, may an Avel receive an Aliyah?
May an Avel during Shiva be a Gabbai and give out the Aliyos?
Some write that he is to avoid doing so.
May an Avel during Shiva perform Hagbah/Gelila/Pesicha?
Sitting down: If the Avel does Hagbah he may sit down afterwards on a chair with the Sefer Torah.
 Michaber Y.D. 384:2; Tur 384 in name of Maharitz Geios; See Nitei Gavriel 97 and 116; Pnei Baruch 10; 23:23; Ketzos Hashulchan 85:10; Piskeiy Teshuvos 135:21; Yalkut Dinei Kerias Hatorah p. 1342
Custom of Rebbeim: The Rebbe Rayatz received an Aliyah on Shabbos and the weekday [i.e. Thursday] during the Aveilus. These Aliyos were not regular Aliyos that the Rebbe received weekly. [Igros Kodesh 6:103; Reshimos Hayoman p. 413, printed in Shulchan Menachem 5:270] The Rebbe ibid concludes that since he has not found a reason for the above, therefore he only received an Aliyah on Shabbos during Shiva. [Igros Kodesh ibid] However, in 1965 during the Shiva for his mother, the Rebbe received an Aliyah on a weekday [i.e. Monday] during the Shiva. [Shulchan Menachem ibid footnote 18]
 The reason: As it is forbidden for an Avel to learn Torah. Now, although an Avel is obligated to hear the Torah reading, nevertheless it is forbidden for him to get an Aliyah, as he may not teach Torah to others, and in previous times the custom was for the person getting the Aliyah to read the Torah out loud. [See Nitei Gavriel 97 footnote 6] Vetzaruch Iyun from M”B 548:16
 Beir Hagoleh ibid
 Y.D. 400:1
 Rama ibid; Shach 384:3; Hagahos Maimanis 3
The reason: As otherwise it would be public Aveilus which is forbidden on Shabbos. [Shach ibid]
 Hagahos Maimanis in name of Maharam; Chasam Sofer 352, brought in Pischeiy Teshuvah 401:3; Poskim brought in Nitei Gavriel 116:2 footnote 3; Ketzos Hashulchan 85:10
 Some Poskim rule the Levi is to also leave the Shul. [Aruch Hashulchan 400:9] Other Poskim rule the Levi may receive the Aliyah. [Minchas Yitzchak 9:130] See Nitei Gavriel 116:3 footnote 5
 Ketzos Hashulchan 85:10
 Michaber ibid
The reason: As if he does not go up it would be public Aveilus which is forbidden on Shabbos. [Michaber ibid]
 Michaber ibid that so did Rabbeinu Tam when he sat Shiva. Rabbeinu Tam was accustomed to always receive Shelishi, and during his Shiva he went up on his own for the Aliyah of Shelishi, even though the Gabbai did not call up him up. He said the reason he did this is because since he receives Shelishi every Shabbos, one who now sees that he is not going up for Shelishi will say that he is not going up due to his Aveilus, and it is forbidden to show public Aveilus on Shabbos. [Michaber ibid]
 Custom of Rebbe and Rebbe Rayatz, brought in Igros Kodesh 6:103, printed in Shulchan Menachem 5:270
 Meiri Moed Katan 15a; Nitei Gavriel 97:3
 M”B 135:45 in name of Elya Raba and other Poskim; Shaarei Efraim 8:9
 Shaarei Efraim 8:9; Derech Hachaim
 See Nitei Gavriel 97 footnote 8 in name of Tiferes Yisrael Gittin 5:56
 Taz 400:1 in name of Rashal 71; Maharit; Kneses Hagedola, Nachalas Shiva 17; Neman Shmuel 39, brought in Birkeiy Yosef 400; Poskim brought in Nitei Gavriel 116 footnote 11 and 136:6
 Peri Chadash; Daas Eish 6 brought in Pischeiy Teshuvah 400:5
 The reason: As otherwise it is considered public Aveilus. [ibid]
 Taz 402:5 based on Rabbeinu Yechial in Tur 402; Gilyon Maharsha 384:1
Other opinions: Some Poskim rule he may not receive an Aliyah even on Mincha of Shabbos, as we do not say Miktzas Hayom Kekulo on Shabbos. [Bach, brought in Taz ibid; The Taz negates his opinion]
 Pischeiy Teshuvos 399:1 in name of Eish Daas; See Nitei Gavriel Sukkos 101:5
 See M”B 548:16
 Nitei Gavriel 97:6
 Nitei Gavriel 97:11
 Taz 384:1; Aruch Hashulchan 384:8
 Derisha 384; Pnei Baruch 10:24