Would there be a problem with leaving a recorder on from before Shabbat or Yom Tov, and thereby recording a lecture given on Shabbat itself? There is a wealth of literature on the subject of Melacha that extends into Shabbat, but I haven't found this particular issue addressed. I have also heard from reliable sources that R' Shlomo Zalman permitted this, but I have yet to find this source inside. Can anyone give me some direction?
Recording on shabbat is linked to use of microphones on Shabbat (since you require microphones to record). It might be in this context that you heard R Shlomo Zalman Auerbach (RSZA) being quoted since he wrote on that topic.
The following is an excerpt from a an article by RSZA in 5706 (1945-6) on the subject of radio broadcasts on Shabbat and Yom Tov. In it, he raises all the relevant problems regarding electricity on Shabbat, and rejects them one by one.
[These prohibitions] apply only to creating or breaking a circuit in a fan, refrigerator, etc. However, when one speaks into the microphone of a radio, no sparks are created, since the manufacturers [of the microphone] are very careful to avoid this. Therefore, even if there is a malfunction and a spark is created, no transgression is committed since it is merely a davar she-eino mitkaven (an unintended result). Thus, when one speaks into a [broadcast] microphone on Shabbat, he does not cause an act of lighting or extinguishing [a flame], or any other forbidden melacha. Rather, the speaker merely causes a change in current which affects the radio waves being broadcast, such that the membrane of the radio receiver vibrates in accordance with the sound waves of the speaker. It would therefore seem that there is no need whatsoever to be concerned with the problems of makkeh b’patish, tikkun mana, or molid, since his speech does not cause any Shabbat prohibition to be violated.
However further on (section 4) R Rozen writes Rav Auerbach leans towards prohibiting microphones (see there for details).
On the Igrot Moshe cited in another answer on this page R Rozen writes
I have not merited to understand Rav Feinstein’s reasoning here, and he himself wrote that the matter was not entirely clear to him. It is true that speech affects the strength of the current in the microphone, and particularly in the amplifier, but this is merely a change in the current, not the creation of one. There is no closing of a circuit involved, nor any hav’ara, nor completion of a task, nor any lasting impression; I therefore see no room for his suspicions. Furthermore, we will later quote from Rav Feinstein’s own ruling on hearing aids, where he states that the suspicions of a prohibition are unclear, and therefore in cases of great need may be permissible.
Many poskim forbid the use of microphones on Shabbat, i.e., R Eliezer Melamed (partly because of marit ayin) even if Zomet and others believe it is permissible. For more sources see also here and there.
Recording is even more complex since part of the reason to permit microphones is the great need from some to hear - and that need might not exist when recording audio.
It is forbidden to speak into a tape recorder even if the recorder was turned on before Shabbat
Igrot Moshe OC 3:55 forbade recording a shiur on Shabbat or the like since speaking into it might cause the voltage to increase in the circuit and also it might be considered molid to have something new introduced into the memory of the recorder. See also Menuchat Ahava 24:13, Yechave Da'at 2:57 (Halachipedia)
Though I'm still not sure what R' Shlomo Zalman said about it.