It depends on the meters and what battery you use as a replacement.
Zinc-Air "hearing aid" cells have identical characteristics, but run flat all on their own within months or even weeks - apart from their short life (and the lack of big, high capacity versions) these are perfect plug in replacements.
Silver Oxide have 0.2V higher voltage, but a fairly similar discharge pattern - with some meters that creates a negligible constant error below one stop, on others it will still be within the range that can be compensated on the sensitivity dial. But in some cases, the resultant offset goes past the (usually limited) range of sensitivities, and you will need to insert extra electronics (a drop diode) to make it work.
Alkalines have very different, non-linear discharge characteristics, so that measurements at various discharge stages need a different correction - making a circuit designed for mercury or silver cells work with alkalines usually requires significant rebuilding of the electronics.