The energy saved by a new washer or dryer vs an older one is dwarfed by the energy needed to make the new washer or dryer.
FWIW, the same applies to most building replacements. Yes, the newer buildings use less energy, but the savings doesn't pay for the cost of replacing an old one.
This really depends on what you're replacing. Replacing a standard electric water heater that uses ~$500/year of electricity with a heat pump water heater that uses ~$100/year (that's about what ours uses) has a pretty quick payback period.
FWIW, the same applies to most building replacements. Yes, the newer buildings use less energy, but the savings doesn't pay for the cost of replacing an old one.