Related, among others, is "Why did they get rid of Carson and Weir?" on GateWorld.net.
As when I asked about Why was Crusher replaced with Pulaski who was replaced by Crusher? I have found that on these types of questions there is a lot of discussion in forums around the time of the event, but no definite source of all the tidbits over time which come out.
With the spectacles of history on our side, what are the reasons Torri Higginson and Dr. Weir left Stargate: Atlantis?
It seems to have been a combination of three factors:
Elizabeth Weir was never intended to be a main character; she was intended to be the support character, specifically to be like SG-1's General Hammond. However, she seems to have been a personal favourite of Atlantis' co-creator Brad Wright.
Joseph Mallozzi, the executive producer of Atlantis starting in season 2, said in a January 2008 blog post that Wright was one of the big reasons Weir became as major of a character as she did (emphasis mine):
Weir’s role was originally envisioned as a parallel to the Hammond role on SG-1. And so, unlike the team, it wasn't intended that she head off-world every episode. Like it or not, the character of Weir was intended to play a supporting role to that of the team. Still – while Brad was running Atlantis, he was very supportive and protective of the character, working hard to make sure she got her fair share of stories, and to suggest otherwise is not only disingenuous, but a slap in the face to a guy who was the driving force behind episodes like "Before I Sleep", "The Real World", and "The Long Goodbye".
Despite that, Higginson's been pretty clear that she wasn't very happy with the quality of writing she was getting. According to a partial transcript of a 2008 livechat:
Torri was always trying to bring more to Weir during those pauses. She was frustrated with how Weir was being written; she wasn't being active, instead was passive, so how could she be a leader.
And in an interview with GateWorld.net she said:
I think that my character -- we never found a place for her. I think everybody can take a bit of responsibility for that, obviously myself included. So yeah, I imagine that was it. They thought "here's this woman character that we're not really able to explore to her full right."
So many of the episodes I was just there in the background, which wasn't challenging for me. And I think they're going "Why are we paying this chick when she's only in for a couple of scenes?" You know what I mean? It made a lot of sense, I thought. That's how I hypothesize it. It makes a lot of sense for me.
So a decision came down following season 3, coinciding with Wright's own role in the series diminishing, to reduce Weir's character dramatically. Originally the producers wanted to make Weir a recurring character, but Higginson instead chose to leave the show. In the livechat I linked to above, Higginson recalled the story:
But I found out, because I kept going to them, I kept going up to them saying 'I have a feeling my character, you’re not doing anything with me, and you guys have me for six years and I don't want to, you know, be here not doing anything. Let me know what's going on.' And they kept saying "no, no, no, it's great. We love you. We love you. Things can be great.' And I said 'well, if that’s the case, can we do something with her' and they kept reassuring me that nothing - and the very last day of filming season three, as I finished filming the last scene on the last day I was called up to the office and was told that my character was going to become recurring if I chose to be. So, I thought that was not very, um, dignified, way to deal with it, and I was a bit surprised. So I was— so my reaction was one of yeah, I was a little bit surprised. I was a little bit upset by how it was dealt with. But I wasn't upset at the decision because I understood it. I kept going to them saying 'I get, I get what's going to happen, just give me some notice so I can pack my apartment and move back to L.A. Really. So I wasn't upset with the decision. I was upset with how it was handled."
They wanted Samantha Carter (Amanda Tapping) so Weir had to go to make room for her. Atlantis' ratings were lower than SG1's and they thought importing her character might help. Unfortunately for them Tapping had another project in the works, Sanctuary, which ended up drawing her away.
Not that I don't like Carter but I think ditching Weir was a mistake. She was a great character.
And they seem to have been a bit rude in how they handled it. They said they didn't tell her until after the final day of shooting in case she responded badly, i.e. they treated her as if they thought she was unprofessional. I would have liked to see her character come back even in a limited recurring fashion but I can understand why she declined that offer.
Atlantis' ratings were lower than SG1's and they thought importing her character might help. Interesting, but if you can provide a source for this information, it would greatly improve the quality of this answer.