How to use a variable to indicate a file descriptor in bash?

You have to use eval and put the entire expression in quotes.

eval "exec $id<>$file"

And do that every time you want to use $id.

The accepted answer is correct, but as of bash 4.1, you can use automatic file descriptor allocation, and in that case you don't need eval:

file=a
exec {id}<>"$file"

Then you can use it like this:

echo  test >&${id}

or:

fsck -v -f -C ${id} /dev/something