Does the MAPQ=0 fraction of a BAM file depend on the insert sizes? When doing Illumina 2x150bp sequencing of genomic DNA, and after aligning the reads to GRCh38, does the percentage of the non-N fraction of the human genome as MAPQ=0 depend on the insert sizes of the genomic fragments?

This is, for two identical samples with identical final coverage, sample A having an average insert size of 250bp and sample B having an average of 450bp, would the fraction of MAPQ=0 change between the two?
Yes, as a general rule of thumb mappability increases with insert size (up to a limit) and read length. Whether this will actually occur in a given case will depend more on how randomly the sequencing samples from the genome to begin with (i.e., if the library prep happens to select for/against high mappability regions then the insert size won't much matter). The basic reason behind this is that if one end of a read aligns to a repetitive element that the likelihood of being able to use the other end as an anchor will increase a bit if it's further away. Now having said that, Illumina sequencers in particular like a certain fragment length for efficient sequencing, so you can't go too crazy on increasing the insert sizes and still get decent output (unless you modify your library prep).