I think the question title says it all, but is there a reason that I should or should not put a surge protector on my washer or dryer? Growing up, I don't recall ever seeing them in my parents' house nor at laundromats. But given that this is a several hundred dollar appliance, I'm not sure why these aren't protected by a surge protector.
If they have the simple electromechanical timers, there's not much that can be damaged by 'reasonable' surges - damage from a lightning strike on your line wouldn't be stopped by a surge protector.
Electronic controls typically do have a surge protector built onto the board, so there's no need for an external one.
A surge protector can protect from surges from:
- Motors or other large devices
- Incoming power problems
A washer or dryer is actually more likely to cause the first category of problems than to be harmed by it. That still leaves open the question of lightning or other external problems. As a general rule, the protection from a surge protector is primarily for sensitive electronics. Historically, washers and dryers had very simple electromechanical timers. Many newer appliances do have sophisticated electronics - e.g., if your appliance can communicate via WiFi then it has, essentially, a computer inside that could use protection.
However, it is also quite possible that the heavy electrical loads, particularly from an electric dryer, may stress the protective electronics in a surge protector quite a bit. I certainly wouldn't put a washer or dryer on a typical $10 - $25 basic surge protector. (Actually, for an electric dryer that wouldn't even be an option, but for the major appliances where it is possible, I still wouldn't do it.)
I would consider a whole-house surge protector as an option. I am skeptical (most of my experience being with the very logical "load plugged directly into a surge protector") of a whole-house surge protector providing the same level of protection as point-of-use surge protectors, but it should still provide significant protection, and do so for everything in the house, including major appliances.
First, if by "surge suppressor you mean power strip or some sort of plug-in dingus, those kinds of cheapies are Right Out. They are way too flimsy to carry the power for a major appliance. They also shouldn't be used with air conditioners, microwaves or any heat making appliance.
They do make some commercial-grade plug-ins (e.g. Tripp-Lite), and you can also use units intended for "whole house surge suppression" on a major appliance, simply by locating it near the appliance instead of at the main panel.
You should know that electric dryers have only a big resistor across the 240V poles. Their motors and electronics sit between one of the poles and neutral. That is the only place you really need a surge suppressor.
It can be a good idea to put a surge suppressor on appliances which are a source of noise and spikes. A washer and dryer are definitely two of them.