Gemara

(R. Avina): Anyone who says "Tehilah l'David" three times a day is guaranteed to be a Ben Olam ha'Ba. (He does not need punishments before getting a share in the world to come.)

Question: What is the reason?

Suggestion: This is because it is written according to the Alef-Beis. (It declares Hash-m's praise in a full manner.)

Rejection: If so, the same should apply to "Ashrei Temimei Derech" (Tehilim 119), which is written according to an eight-fold Alef-Beis!

Answer #1: It is due to the verse "Pose'ach Es Yadecha..." (Hash-m feeds the entire world.)

Rejection: If so, the same should apply to Hallel ha'Gadol (Tehilim 136), which includes "Nosen Lechem l'Chol Basar"!

Answer #2: It is due to both of these attributes.

31a (Beraisa): One may not start to pray amidst sadness, laziness... rather, amidst the Simchah of a Mitzvah.

32b (Mishnah): Chasidim of old would wait one hour (before praying).

(R. Yehoshua ben Levi, and a Beraisa): We learn from "Ashrei Yoshvei Veisecha."

Rishonim

R. Yonah (23a, DH Kol): The primary reason we say Tehilah l'David in Pesukei d'Zimrah is due to the verse Pose'ach Es Yadecha. Therefore, the Ge'onim say that one must have intent when saying it. If he did not have intent, he repeats it. It seems that even if he did not intend in all of Tehilah l'David, if he intended in this verse, it suffices.

R. Bechaye (Vayikra 7:37, cited in Magen Avraham 1:7): Chachamim expounded that anyone who engages in Parshas Olah (or Minchah or Chatas...), it is as if he offered one. This is when he contemplates the meaning of the Parshah and what it hints at. Through this he will understand wonders of the laws of Korbanos. This will lead him to try harder to fulfill Torah and Mitzvos, and his sins will be forgiven as if he offered a Korban. Chazal did not discuss one who will merely say the verses without contemplating their meaning. 'Anyone who says Tehilah l'David three times a day is guaranteed to be a Ben Olam ha'Ba' is similar. Through contemplating what the verses teach and what they hint to, he will recognize the wonders of the All-knowing, and his heart will be strengthened in Emunah is Hash-m and His service. Through this, he will inherit the world to come.

Machatzis ha'Shekel (on Magen Avraham 1:7, b'Sof): Through saying Tehilah l'David with intent, he will recognize some of Hash-m's Chesed and greatness. This will make him zealous to fulfill Hash-m's word and guard himself from sin. If he sinned, he will repent fully. Through this, he merits the world to come.

Tosfos (31a DH Rabanan): We enter Tefilah only amidst seriousness and Simchah of Mitzvah, e.g. engaging in Torah. Therefore, the custom is to say Pesukei d'Zimrah and Ashrei before Tefilah.

Poskim

Shulchan Aruch (OC 51:7): One must have intent in the verse Pose'ach Es Yadecha. If he did not, he repeats it.

Beis Yosef (DH v'Tzarich): Ohel Mo'ed says that saying Tehilah l'David three times a day makes one a Ben Olam ha'Ba, i.e. this Mitzvah can be decisive, but it alone does not make him a Ben Olam ha'Ba.

Magen Avraham (6): Nowadays we do not repeat (Shemoneh Esre) due to lack of Kavanah (101:1). There, we are concerned lest he not have Kavanah also the second time. Here, it is only one verse. He repeats it, for he can have intent the second time.

Kaf ha'Chayim (32): Also, we do not repeat Shemoneh Esre, for if he does not intend in the repetition, also those Berachos were l'Vatalah. This is not a concern here. Also the Birkei Yosef says that he needs to repeat only the verse Pose'ach Es Yadecha, unlike the Levush.

Mishnah Berurah (16): The Chayei Adam brings from the Levush that one must say from Pose'ach Es Yadecha until the end in order. If he did not remember (that he did not have Kavanah) until he already said other Mizmorim, and he does not have time to return (to say from Pose'ach Es Yadecha until the end), in any case after the Tefilah he says from Pose'ach Es Yadecha until the end.

Shiyurei Berachah (of the Birkei Yosef, 3, citing Rav Amram Gaon): One must say Tehilah l'David once a day. Later Chachamim enacted to say it in Pesukei d'Zimrah, before u'Va l'Tziyon and before Minchah, lest he be negligent once or twice, so in any case he will say it at least once. A proof is from Shabbos, in which we say Ashrei only twice. In Bavel they do not say it before u'Va l'Tziyon. They learn from Shabbos.

Shiyurei Berachah: Nowadays everyone says it three times a day, even on Shabbos.

Kaf ha'Chayim (33): Tif'eres Shmuel says that it was enacted to say it three times a day, so at least once he will say it with Kavanah and merit the world to come. Ma'amar (Mordechai? - PF) says that their (Tif'eres Shmuel's and the Birkei Yosef's) text of the Gemara said 'anyone who says Tehilah l'David every day is guaranteed to be a Ben Olam ha'Ba.' If so, if one said Pose'ach Es Yadecha (in Pesukei d'Zimrah) without intent, and remembered only after he said other verses, he need not repeat it, for he has compensation later. Why should he say it not in its proper place? However, if he said Pose'ach Es Yadecha in Tehilah l'David after the Amidah without intent, he repeats it, for there is no concern for a Hefsek.

Rema: We say "v'Anachu Nevarech Kah..." at the end of Tehilah l'David.

Beis Yosef (DH v'Nahagu): The custom is to say Ashrei Yoshvei... before Tehilah l'David, for we learn from it that one must delay for a Sha'ah before praying.

Beis Yosef (DH v'Acharav,citing Kolbo 4): The Eshkol says that the custom is to say "v'Anachu Nevarech Kah..." after Tehilah l'David. Maran ha'Levi explained that one who says it three times a day is a Ben Olam ha'Ba. Therefore, after saying it, we say that we will always bless Hash-m, i.e. we will merit the world to come. I say not to add the verse. David did not say it. Also, whatever Parshah was dear to David, he began and ended with Ashrei (10b). We can similarly say that (Tehilah l'David was dear to him, for) he began Tehilah l'David and ended Tehilas Hash-m... Therefore, we should not add to it.

Beis Yosef (ibid.): Even though according to the Eshkol we should add the verse only in Pesukei d'Zimrah, like Orchos Chayim says, once they enacted to add the verse, we always add it.

Taz (3): Rav Amram Gaon says that we add v'Anachu... because from here until the end of Tehilim, every psalm ends with Hallelukah, and those psalms begin after Tehilah l'David with Hallelukah.

Teshuvas Maharshal (64): When I reach Ashrei, I skip the first two verses that begin with Ashrei and begin Tehilah l'David. The verse was appended only in order to delay a Sha'ah before Tefilah. Therefore, I say it only at Minchah. I skip the second verse "Ashrei ha'Am...", like Tosfos says in Berachos.

Hagahah: It seems that 'like Tosfos says in Berachos' belongs above, to explain why we say Ashrei Yoshvei... before Minchah.

Maharshal: Only the first verse Ashrei was enacted beforehand. It ends Yehallelucha Selah, and afterwards we say Tehilah l'David. Afterwards, I add afterwards v'Anachu..., even though (at Minchah) the idea of linking Hallelukos does not apply. Since one began with Yehallelu, it is proper to end with Hallelukah. Before la'Menatze'ach, I begin with Tehilah l'David and end with Tehilas Hash-m. V'Anachu applies only in Pesukei d'Zimrah, when we link Hallelukos. Even though some always add it, I found some who say to begin and end with Tehilah.

Magen Avraham (7): The Maharshal said that we add v'Anachu... in order to finish with Hallelukah, to match the beginning of the next psalm. Therefore, we do not say it at Minchah, and in Shacharis we do not add Ashrei Yoshvei... We begin with Tehilah l'David. However, the custom is to say v'Anachu also at Minchah.

Rebuttal (R. Akiva Eiger): The Maharshal did not say not to say v'Anachu at Minchah! Rather, he said that we say Ashrei Yoshvei only at Minchah, to delay before Tefilah. In Shacharis (Pesukei d'Zimrah) we add v'Anachu only to link Hallelukos. This does not apply at Minchah, but then we add this verse since we began with Ashrei... Yehallelucha, so it is proper to end with Hallelukah. Before la'Menatze'ach, he does not add anything.