Generically via bedmap:

bedmap --echo --echo-map --count exome.bed annotations.bed > answer.bed

The annotations.bed file can be GENCODE genes, OMIM genes, etc. which have CDSs, exons, etc.

For example, GENCODE gene counts can be obtained via convert2bed:

wget -qO- https://ftp.ebi.ac.uk/pub/databases/gencode/Gencode_human/release_44/gencode.v44.annotation.gff3.gz \
    | gunzip --stdout - \
    | awk '$3 == "gene"' - \
    | convert2bed -i gff - \
    > genes.bed

Then:

bedmap --echo --echo-map-id-uniq --count exome.bed genes.bed > answer.bed

The last column of answer.bed will have the number of genes that overlap an entry in exome.bed.

You can repeat this for other sets and modify awk statements etc. to get specific subsets of annotations.

OMIM annotations may need an intermediate map step to go from MIM numbers to ENSG to bedmap back to BED entries.