In the opening crawl, the Jedi Knights are described as "the guardians of peace and justice in the galaxy," which is taken verbatim from Obi-Wan's dialogue in A New Hope: "For over a thousand generations, the Jedi Knights were the guardians of peace and justice in the old Republic before the dark times, before the Empire" [33:54].
3:05 - Obi-Wan: "I have a bad feeling about this."
5:02 - Nute Gunray (after Sidious commands him to kill the Jedi): "Yes, yes, my lord, uh, as you wish." The phrase "as you wish" appears a number of times in the original trilogy: in A New Hope, Vader says it after Tarkin asks him to release General Motti from his Force choke [38:45]; in The Empire Strikes Back, Boba Fett says it after Vader warns the bounty hunters against disintegrations [1:06:22]; in Return of the Jedi, Vader says it twice more in conversations with the Emperor [39:16 and 1:07:38], and Admiral Piett says it to Vader ("As you wish, my lord") [56:13].
6:27 - Nute Gunray: "Close the blast doors!" A stormtrooper also says this in A New Hope [1:29:39].
23:25 - Captain Panaka (referring to the battle droids): "There are too many of them." A rebel pilot says basically the same thing in Return of the Jedi (referring to TIE fighters): "There's too many of them" [1:35:13].
23:48 - Sio Bibble: "Our only hope is for the senate to side with us." Leia uses the phrase "only hope" twice in A New Hope ("Help me, Obi-Wan Kenobi; you're my only hope" [35:35] and "You must use the information in this R2 unit to help plan the attack; it's our only hope" [1:38:49]); C-3PO uses it once in The Empire Strikes Back ("Your highness, we must take this last transport; it's our only hope" [31:49]); Obi-Wan uses it once, referring to Luke, in Return of the Jedi ("You were our only hope" [47:24]); and Luke uses it once, referring to Leia, in Return of the Jedi ("If I don't make it back, you're the only hope for the Alliance" [1:19:39]).
31:47 - Qui-Gon: "I need parts for a J-type 327 Nubian."
35:30 - Qui-Gon: "I'm sure another solution will present itself." Perhaps coincidentally, this is somewhat similar to what Dr. Jones, Sr. says in The Last Crusade: "I find that if I just sit down and think, the solution presents itself" [1:02:16].
39:03 (to Padmé) & 39:26 (to R2-D2) - C-3PO: "I am C-3PO, human-cyborg relations." He introduces himself the same way to Luke in A New Hope ("And I am C-3PO, human-cyborg relations" [20:25]) and to Lando in The Empire Strikes Back ("I am C-3PO, human-cyborg relations" [1:21:31]). He's about to say the same thing in Return of the Jedi, but EV-9D9 interrupts him, so he gets only as far as "I am C-3PO human-cy-" [11:21].
56:08 - Podrace announcer: "And back again it's the mighty Dud Bolt with that incredible racing machine, the Vulptereen 327."
1:17:31 - Anakin (to Qui-Gon): "Are you alright?"
and
1:19:06 - Padmé (to Anakin): "You alright?"
In A New Hope, Luke asks Obi-Wan, "Are you alright?" [59:16] and Biggs asks Luke [1:46:44], and in The Empire Strikes Back, Luke asks, "Rogue Two, are you alright?" [30:31]. Twice in The Empire Strikes Back, Han asks this with the shortened "You alright?" once to Luke [22:12] and once to Leia [31:39].
1:28:36 - Queen Amidala: "I was not elected to watch my people suffer and die while you discuss this invasion in a committee." Han Solo uses a similar expression in The Empire Strikes Back: "No time to discuss this in a committee!" [1:00:16].
1:33:06 - Ki-Adi Mundi (referring to Anakin): "The Force is strong with him." In The Empire Strikes Back, Palpatine, speaking to Vader, also says this about Luke [53:46]. Vader has a similar line in A New Hope when he says, "The Force is strong with this one" [1:55:41], also describing Luke, although unknowingly.
1:34:50 - Obi-Wan: "It's not disrespect, Master; it's the truth."
Qui-Gon: "From your point of view."
Qui-Gon's line resembles what Obi-Wan says to Luke in Return of the Jedi: "So what I told you was true, from a certain point of view" [46:21].
1:36:33 - Qui-Gon (to Queen Amidala, after she says that Senator Palpatine fears the Trade Federation wants to destroy her): "I assure you I will not allow that to happen." The context is quite different, but Palpatine (as Emperor) also uses this expression while speaking to Luke in Return of the Jedi: "I assure you we are quite safe from your friends here" [1:32:08].
1:37:55 - Queen Amidala (to Jar Jar): "I need your help." Obi-Wan also says this in A New Hope: "I need your help, Luke. She [Leia] needs your help." [36:03].
1:47:37 - Wilhelm scream (Naboo guard)
1:59:48 - Obi-Wan screams, "No!" after Darth Maul strikes down Qui-Gon, which is the same sort of situation as in A New Hope when Luke screams, "No!" after Darth Vader strikes down Obi-Wan [1:32:24]
2:00:19 - Battle droid (to Jar Jar and Captain Tarpals): "Hands up!" A storm trooper also says this to Han and Leia in Return of the Jedi [1:48:07].
2:03:31 - The battle droid next to Jar Jar has markings that appear to be 1138:
2:06:41 - There's a brief musical cue of the Imperial March as Yoda tells Obi-Wan that he fears "grave danger" in Anakin's training.