2 of 4 added 530 characters in body

Open the two files on different file descriptors. Redirect the input of the read built-in to the descriptor that the file you want is connected to. In bash/ksh/zsh, you can write read -u 3 instead of read <&3.

while IFS= read -r lineA && IFS= read -r lineB <&3; do
  echo "$lineA"; echo "$lineB"
done <fileA 3<fileB

This snippet stops when the shortest file has been processed. Replace read … && read … by read … || read … if you want to keep processing the longest file ($lineA will be empty if you keep processing fileB and vice versa). Use IFS= read -r lineA; IFS= read -r lineB; if you want to process till the end of fileB.

See also When would you use an additional file descriptor? for additional information on file descriptors, and Why is `while IFS= read` used so often, instead of `IFS=; while read..`? for an explanation of IFS= read -r.