What happens if you do NOT recruit LIARA in ME-1?

Nope, pretty sure she's essential to figuring out the Ilos thing. I have ignored her till the last minute, got her, then told her everything that had happened on the previous 3 planets.
Her shocked reaction was pretty great.

Garrus will mention not being recruited in ME2 so it's doable, not sure about Wrex tho.

Finding and recruiting Liara is part of the main plot, as she is the only one that recognizes the planet Ilos.

As others have said, there is no possible way to bypass recruiting Liara as Ilos will not unlock without her input. Basically, if you do the other 3 (Feros, Noveria, and Virmire), Shepard will say he/she has only one lead left... and that's finding Liara, so then the main plot will not advance further until you go to Therum.

You must recruit one of either Wrex or Garrus.  If you have already recruited the other, you can turn down their offer to join you.  There are two times when you can do this... If you haven't spoken yet to the other at all, you can turn down the first one you talk to, but then you will not be given an option to turn down the other.  You can also opt to completely ignore talking to one or the other prior to taking down Fist, and then you will be approached by that one before getting onto the elevator to the Normandy... and you will be able to turn that one down at that point as well.

As themikefest said... If you don't recruit Wrex or if you killed him in ME1, you will be dealing with Wreav in ME2.  With Garrus in ME2, it really just changes that little bit of introductory dialogue when you does the Archangel mission (since Garrus is a mandatory squad mate in ME2).

I'm pretty sure you need to recruit her . I wish we didn't need to recruit tali as it is a multi choice game but her your forced to recruit squad mates you don't want.

Meh I just leave them in their hidey holes on my ship and never speak to them . ;-)

I usually get her first but most recently she came after Feros.  That was different in itself.  I feel too bad not to have her on Noveria but someday I might give it a shot.

As many have said, her recruitment is mandatory.

In fact, Liara is the toughest character to kill in the entire trilogy.

Miranda is the hardest to kill in ME2 (she can only die at the last segment, either defending the line or going with shepard without being loyal), but also the easiest to kill in ME3 (you have to fulfil a lot of requirements for her to survive).

Liara is the only character from ME1 and 2 assured to be alive during ME3. Which is why she and Edi (and also James) are so important in ME3. Everything related to Tali, Garrus, Kaidan and Ashley is conditional. Thats is why Liara has such a major role in ME3 main story, Citadel, From Ashes. Strangely, James and Edi become more important than her in Leviathan. I guess the writing team was facing way too many accusations of favoring Liara. But come on, she was the one who came up with the theory about several cycles of extinction. It is hard to imagine that shepard is investigating a (quite) ancient creature and does not want to hear her input. Its like dealing with a totally strange disease and not wanting to consult mordin.

The only way to having the death of Liara is during the last moments of ME3. 1-She can die by picking up the refusal ending. 2-If you have low EMP take Liara with you to the last section of the game and Harbinger will kill her. 3-Do not take her with you. When the normandy crashes landed and no one will come out from the door, which wont open, leading to presume that all hands perished.

If you recruit her last, its the only time when Shepard could say anything with utter skepticism to her. She only knew of Ilos from watching vids about it, not really from her decades of research and studies (on average Asari researchers themselves have centuries of studying and branching the area of their studies...). The Ilos codex itself mentioned other Asari scientists trying to do multiple expedition by using the old fashion FTL to reach Ilos. Why were they never mentioned? and I never get was why she never go back to Ilos to study the place. Its a huge big deal for a Prothean scientists, a planet with Prothean research facilities filled with thousands of dead Prothean corpses and she just put up pictures of the place at her apartment and that's enough? (another pathetic shit about ME2, no one questions why would prothean cryo-stasis themselves enmass for hundreds of years until the planet became their graveyard)

As for questioning why the Protheans put themselves in cryo en masse, that certainly goes toward support of everything Shepard ever said.

You also forget that squadmates don't necessarily tell you everything and sometimes they intentionally choose to omit the entire truth. Liara does have her own research team. Paragon Lost's Treeya Nuwani who is an Asari Anthropologist had Liara as her mentor and contacted her directly for advise. In Citadel DLC, she talk about having colleagues in the prothean sites who happened to teach her piano.

And was it a coincidence that the Asari Councilor knew about Liara was on Therum or that she's being only one on that dig site surrounded with lava with all heavy equipment without any ride home. Its also impossible for even a project to be left completely unattended that way without any others supervising the project... which is irresponsible, unethical... what if bad things happen or she destroyed the site... which did happen... oh crap. 

Was she ever punished for that? Oh.. just make her a persona non grata.... to the only Prothean scientist accompanying the deceased Spectre who rediscovered Mu relay and Ilos, who mindmeld with that Spectre who experience the Prothean visions leading them to Ilos... who knew of the existence of the only other Asari who hold the Prothean cipher. Nah... just drop that Ilos shit around, and concentrate on making Liara the next Shadow Broker who occasionally do Prothean consultations as a side because her doctorate isn't revoked and according to ME3 and MEA, her research stuff is amazeballs and not even a single one talk about the Prothean dig site that got destroyed or that Liara get pwned by Javik who tell her she "knows nuthin".

That was a gross assumption. Being a Matriarch doesn't guarantee anyone's position in the Asari society, it just show you could survive long enough without being a road kill to reach to that stage. The Asari Councillor herself immediately support Shepard's application to the Spectres simply for Benezia association with Saren through a voice clip. That was like signing a death warrant based on speculation of treason and it turns out Benezia was a victim herself. If Benezia simply have her own followers to be notable but so does a lot of Asari Matriarchs. Aetytha have her own asari mercenaries. Samara was attached to her Justicar order. Aria have the entire Terminus System groveling to her. All of them have their connections to the Asari government.

Being born to a matriarch doesn't entitle an Asari to have higher status than anyone. Liara isn't an Asari princess simply for having a Matriarch as a mother, in fact no one else acknowledge her except for being a pureblood. In fact, she's fine with throwing her mom under the bus and disclaiming her association with "Benezia" to the first stranger who asks where her loyalty is. She never even fight to protect her own mother's reputation despite knowing her mom was a victim of indoctrination. That by itself is a perfectly valid personal motivation for Liara herself against the Reapers which somehow none of the writers think of using even once for the entire decade of Mass Effect franchise's existence.

Even if I dislike her, I still think she deserve to be a character who doesn't solely function as a convenient plot device. What next, the Andromeda Initiative turns out to be funded by Shadow Broker? Keri T'vessa being Liara's half-Hanar sister? For a game that tries to cut it ties from the trilogy, they sure as hell hasn't stop their Liara is canon campaign.

*last toss of a coin*

 I did toss out a possibility they wanted impartial investigators. All the notes they ever needed would be in the debriefing after. Whether Anderson was a Councilor or an officer, he was not impartial. The Council would not want those close to Shepard (Liara and Anderson) muddling up the evidence. If Anderson was a Councilor, there were other duties he had to attend to. How would it be if he was running off to Ilos and he was asked why. To support Shepard and confirm Reapers. Since the Council did not advocate the existence of Reapers, it wouldn't do to have the new Councilor adopt a public contrary stance. If he was an officer, the Admiralty Board would pass down the order that he was not to say stuff like Reapers (don't rock the boat, we have what we wanted...at a cost, so shut up).

Since no one had any idea what the Alliance was doing in the two years except definitely rebuilding the chewed up Fifth fleet, it's hard to say. Hackett was just one high rank officer who seemed to cautiously support Shepard, the stance of the rest of the Admiralty Board was murky / not supportive. Hackett was the CO of Fifth Fleet so if he wasn't bounced to other duties, overseeing the rebuilding of the fleet was on his card. If the AB didn't agree with his view that Shepard might be right, they would make sure he stayed close, occupied with that important task. If Anderson was an officer, he could have been roped in to help. It might be an anchor if they're restricted but it was also be an opportunity to prepare the Fifth fleet for battle, incorporating better tech into old and new ships. 

Liara's belief in the Reapers return. A view not widely supported among those who knew of the supposedly Reaper (Sovereign). Not among the Council, unlikely in the Thessian government, even more unlikely in the Alliance government which publicly discounted and played down Shepard's statement after s/he was KIA. Support would be scarce, even among the Normandy's crew who were all deliberately after Shepard's demise. They would be demoralized but she could count on Joker, Ash / Kaidan, Chakwas. Even then, if they had stuck together, there wasn't much they could do. No ship, no big pockets, no reliable sources, either official or unofficial. Even if Liara tried to go to Ilos, she would need to apply to the Council (there was a official sanction research team on Ilos that was evacuated after the Reapers invaded). They would have rejected her; their stance, as I have said before, was a deny of Reapers. They would have told her there was nothing found to support that allegation so there wasn't any need for her to go.

Liara's statement was indicative of her limitations and her resources in the two years. Everyone knew of the SB's ability to find out hard to get information. After trying, I've no doubt, to find a conclusive defense against the Reapers and failing on her own, the SB's network was something she could target. Feron was in his grasp, she could kill off two birds with one stone. The SB's network had existed for nearly a century and no doubt, expanded with the previous SB. The reach, the scope and resources she needed was already in place. After the Bahak System incident, Hackett probably knew there could be no doubt this time, argued with the AB and perhaps convinced them to have Liara at the Mars facility.

*grabs coin, walks away*

End of line.