What effects does multitasking have on comprehension and memory of audiobooks/speech?
There does not seem to be any research on your exact question on:

the effect of &quot;low attention tasks&quot; and &quot;high attention tasks&quot; on the comprehension and memory of speech modality learning.

However, if we widen the scope of the question to different modalities of learning like reading, then there is ample research. However, most of the research seems to be done on comprehension and not memory.
Low Attention Tasks
Lin Lin et. al. (2009) did a study on college students for assessing their reading performance under three conditions: (a) reading only (silence condition), (b) reading with a video playing in the background (background multitasking condition), and (c) reading and watching video simultaneously (test multitasking condition). They found that the participants performed best in the background condition, i.e. when they had a background video playing. This a surprising finding. A similar study was done by Jennifer Lee et. al. (2011) and found that the performance was similar for cases (a) and (b).
For your case, this suggests that &quot;low attention tasks&quot; will not negatively affect your comprehension, and may even increase comprehension (as suggest by Lee et. al. (2009)).
High Attention Tasks
Beyond conclusions about the effect of low-attention tasks, Jennifer et. al. (2011) also rigorously tested the effect of &quot;high attention tasks&quot;. To quote from their abstract:

Participants were given a timed (16 minutes) reading comprehension test in three conditions: Silence (only reading), Background multitasking (reading with a non-tested video shown simultaneously), and Test multitasking (reading with a tested video shown simultaneously) conditions.
Our findings indicated that: (1) participants in the Background condition performed as well as those in the Silence condition, and (2) when participants were tested on their video comprehension, the group in the Test condition performed significantly better than the group in the Background condition. The results of this study suggest that cognitive load plays an important role in determining how much information is retained when students perform more than one task at a time.

Their conclusion tells that the division of cognitive load affects your performance very clearly. Hence, it is best to avoid the high-attention tasks altogether.
References:

Lin Lin, Tip Robertson &amp; Jennifer Lee (2009) Reading Performances Between Novices and Experts in Different Media Multitasking Environments, Computers in the Schools, 26:3, 169-186, DOI: 10.1080/07380560903095162
Jennifer Lee, Lin Lin &amp; Tip Robertson (2012) The impact of media multitasking on learning, Learning, Media and Technology, 37:1, 94-104, DOI: 10.1080/17439884.2010.537664

Non-research Sources
Beyond what the current research says, I would like to cite 2 more sources: &quot;Deep Work&quot; book by Cal Newport, and the training modality of Samurai Jack (a fictional character in an animated series).
In &quot;Deep Work&quot;, Cal Newport makes a very compelling case for why deep work (i.e. work devoid of all distractions) is so important. The first half of the book is completely dedicated to understanding its importance; while the second half deals with its implementation. Cal Newport is an academic, and so he backs his claims with scientific research and his own careful observations.
Finally, as a kid, I was always fascinated with the training regime of Samurai Jack. When training for sword fighting, he used to blind-fold himself and then train. This made him one of the best Samurais, capable of defeating a redoubtable evil dragon. What I got to understand from his regime was that he was blocking all the various stimuli in order to bring intense focus and really bring out the potential of the remaining active senses.
This is not just true for Samurai Jack. Masters in any field have been doing this through the ages : cutting down stimuli.
It's surprising, then, today we are instead adding more stimuli instead of reducing them. I think it's best to go the old-fashioned way. It's best we embrace the The Samurai Jack Blindfold: