Why is Moshe name not mentioned in parshas Tetzaveh?
Me'or Einayim points out that Tetzaveh is (nearly) always read immediately before or after the seventh of Adar, which is Moshe's (birthday and) yahrtzeit. So his name is missing from this parshah specifically, in mourning for our loss.
On a more positive note, there is a long talk by the Lubavitcher Rebbe zt"l (Likkutei Sichos, vol. 21, pp. 173ff; adapted into English at Chabad.org) in which he points out that in a sense Moshe is present even more in this parshah than usual. Its very first word, ואתה, "and you," expresses his existence even more strongly than the use of his name.
He then goes on to explain what Moshe was trying to accomplish with his "threat." He was demanding that G-d be willing to re-establish His relationship with the Jewish People on an essential level that transcends Torah, and that if He wouldn't be willing to do so, then Moshe was prepared to forfeit his place in the Torah in favor of retaining his connection with his people. Thus,
Tetzaveh stands as an eternal tribute to Moses. It is the Torah’s own testimony to Moses’ greatness in relinquishing everything—including his bond with Torah—in order to preserve his bond with his people and restore them to their G-d.
Baal HaTurim says that it was because Moshe said מְחֵנִי נָא מִסִּפְרְךָ אֲשֶׁר כָּתָבְתָּ (erase me now from Your book which You have written) in Parshas Ki Sisa. The question remains why Parshas Tetzaveh was chosen to erase his name. HaRav Shaul M'Vilna answers since Moshe said אֲשֶׁר כָּתָבְתָּ, past tense - therefore it was omitted in Parshas Tetzaveh, the parsha which comes before Ki Sisa. Another reason is that the Gematria of Asher is 501 which is also the Gematria of Tetzaveh.
The Gra says that since most years Zayin Adar - 7 Adar comes out in Parshas Tetzaveh - Hashem who knows what is going to happen left his name out of this Parsha.
In the Sefer Panayach Raza he mentions that since Moshe was originally supposed to be the Kohain Gadol, however he lost this privilege because he said to Hashem at the burning bush שְׁלַח-נָא בְּיַד-תִּשְׁלָח (send, I pray You, by the hand of him whom You will send) therefore in Parshas Tetzaveh which talks about the priestly garments he is not mentioned.
Source for all the above: Torah L'Daas Chelek 1
I have heard in the name of the Ben Ish Chai that Moshe requested Michaini Na Misfracha. The word Misifracha can also read Mesefer Chaf - the 20th Sefer. Since Tetzave is the 20th Parsha therefore Hashem left it out of Tetzave as per Moshe's request.
I do remember hearing that the reason is that Hashem did not want to erase Moshe's name, however since he requested it Hashem kept pushing it off until there was no more pushing it off, and therefore it is not mentioned in Parshas Tetzaveh.
Rav Ovadia Yosef said a great pshat.Moshe Rabbeinu says מחני נא מספרך
לב. וְעַתָּה אִם תִּשָּׂא חַטָּאתָם וְאִם אַיִן מְחֵנִי נָא מִסִּפְרְךָ אֲשֶׁר כָּתָבְתָּ:
He says one could read it מחני נא מספר כ.which means erase me from sefer kaf which equals 20 and the 20th parsha is tzaveh so Moshes plea was granted.
On a peshat level:
A parsha (actually a sidra) is the portion set up to read in a particular Shabbat, as set up in Bavel. In Eretz Yisrael, they had a different division, of a third the size (called a parsha).
I would therefore assume that in Eretz Yisrael, there were a good many "parshiyot" that did not have Moshe's name, and so, Tetzaveh was in no way unique. (Aside from the fact that Moshe was not mentioned at all in sefer Bereishit.)
The reason Moshe does not appear in this parsha has to do
- with topic, in that it is descriptive of the construction of the keilim of the Mishkan and the priestly garments, such that we wouldn't expect Moshe to appear taking action.
- with the number of introductory statements and thus length of the command, where a fewer number of introductory statement will decrease the likelihood of Vayomer Hashem el Moshe leimor
- with the overall number of pesukim is only 101, which is shorter than average.
Of course, this creates an interesting phenomenon which can be used for the sake of derash of derush. And the other answers here cover that. The question, however, assumes that there must be something meaningful about Moshe's absence, and so I think it a good idea to point out that this is an assumption.
The sefer נחל קדומים brings from the sefer מגלה עמוקות that even though Moshe's name is not mentioned explicitly, however, he and two of his previous incarnations are alluded to by the last word in parshas Terumah and the first word of parshas Tetzaveh - נחשת ואתה which is נח, שת, ואתה - Noach, Seth, (two previous incarnations of Moshe) and you (Moshe).
I have heard (sorry for not having an actual reference) from prominent rabbonim that one of the reasons that Moshe Rabbeinu's name is not mentioned in Tetzaveh is due to his plea to Hashem on behalf of the Jewish people after the chet HaEigel (sin of the golden calf):
וְעַתָּה אִם תִּשָּׂא חַטָּאתָם וְאִם אַיִן מְחֵנִי נָא מִסִּפְרְךָ אֲשֶׁר כָּתָבְתָּ
And now, if You forgive their sin... But if not, erase me now from Your book, which You have written."
Hashem forgave the sin, but removed Moshe from the Torah in parashas Tetzaveh.