First day: On the first day of mourning, which is the day of burial, it is forbidden for a mourner to put on Tefillin. [Many Poskim however rule one is to wear the Tefillin without a blessing on the first day of mourning, and so is the Chabad custom. The Chabad custom is to do so even in public. The Chabad custom is to wear even Tefillin of Rabbeinu Tam. According to Chabad custom, all this applies even if the day of death and the day of burial are the same day.]
Second day: On the second day he is obligated to wear Tefillin. [The mourner is obligated to remove his mind from the sadness, and focus on his Tefillin, while he is wearing them.] Nevertheless he is to only put them on after sunrise. Likewise, if new people came to give comfort [Panim Chadashos], which is people that were not there the previous day, then he is not to place them on in front of them, and is rather to wait until they leave. Nevertheless, in the event that he already put them on and then the new arrivals came, he is not required to remove the Tefillin.
If the person was buried at night does one wear Tefillin the next day?
No. However those that are accustomed to wear Tefillin on the first day of mourning, as is the Chabad custom, then the Tefillin are worn without a blessing. Some Poskim however suggest that even a blessing may be recited.
 38/5; Michaber 38/5; Michaber Yoreh Deah 388/1-2; Moed Katan 21a; Sukkah 25b; See Piskeiy Teshuvos 38/6; Chikrei Minhagim 1/265
 Irrelevant to when was the day of death. However see other Poskim in next fotonote
 Admur ibid; Michaber ibid; M”B 38/16; Kitzur SHU”A 211/2; Poskim brought in Piskeiy Teshuvos 38/6 footnote 34
The reason: As the Tefillin are called “Pear/Beauty” as the verse states “Pearcha Chavush Alecha”, and a mourner is wallowing within the dust of his forehead, and it is not beauty to place an ornament under ash. [Admur ibid; Tosafus Brachos 11a]
Other opinions-If the day of death is not the day of burial: Some Poskim rule that if the day of burial is not the day of death then the Tefillin is to be worn with a blessing [Mahartitz brought in Beir Heiytiv 38/4] Based on this many Poskim conclude that one is to place Tefillin without a blessing if it is not the day of death. [See Poskim in Piskeiy Teshuvos 38/6 footnote 34; Chikrei Minhgagim 1/265]
 Mavor Yabok Sefas Emes 34; Poskim ibid regarding if the day of burial is not the day of death
 Rebbe in Toras Menachem 20/202; Hisvadyus 1988 2/585 [printed in Shulchan Menachem 5/271]; Chikrei Minhagim 1/265
 Other opinions: Many Poskim rule one is to wear the Tefillin only in private. [Poskim in Piskeiy Teshuvos 38/6 footnote 34]
 Rebbe in Hisvadyus 1988 2/585
 Based on Mavor Yabok ibid; See Chikrei Minhagim ibid; Mishmeres Shalom 7/2
Other opinions: Even according to those Poskim that rule one is to be lenient to wear the Tefillin on the first day when it is not the day of death, in this case they would all agree it is forbidden to wear Tefillin. [Poskim in Piskeiy Teshuvos 38/6 footnote 34]
 The reason: As the main bitterness [and pain] is only for one day, as the verse states “And their end is like a bitter day.” [Admur ibid]
 Michaber 388/2
 Admur ibid; Michaber ibid
The reason: As by then already some of the time in which one can put on Tefillin has already passed, and we say that Miktzas Hayom Kekulo. [Admur ibid]
Other opinions: Some Poskim rule one may wear the Tefillin on the second day even before sunrise. [Bach, brought in Shach 388/1]
 Admur ibid; M”A 38/5
Chabad custom: Seemingly, according to the Chabad custom to wear the Tefillin even on the first day, these restrictions do not apply. Vetzaruch Iyun, as in this case a blessing is being made.
 Kitzur SHU”A 211/2; Aruch Hashulchan 388/2; Nitei Gavriel 82/5
 Mishmeres Shalom 7/2; Piskeiy Teshuvos 38/6 footnote 35
 See Mishmeres Shalom ibid