I haven't actually been bike touring (no bike here and too low of a travel budget) but this same issue applies equally when hiking and then camping which is what I do on my days off.

I wear contacts, am pretty blind without them (functional, but I sure can't enjoy scenery, and what's the point of hiking then?) and when it comes time to set up camp for the night, my hands are almost always dirty and grubby. if not from sweating or grabbing onto rocks or trees on steep slopes, then from getting the tent set up.

Of course I can't spare the water to get them thoroughly clean (middle eastern summers are no joke), which means there's inevitably some crud on my fingers when I take the contacts out and put them back in. This results in them staying kind of dirty permanently, degrading the quality of my vision.

So...any solutions? Only camp near running water? Get glasses (I've thought about it but sunglasses are also a non-optional part of hiking here, plus the expense is an issue)? Anything else?

I would try some of that orange hand cleaner that you can get at a hardware store. You only need water to rinse it off and it gets your hands very clean. I think it's made from citrus and might be environmentally friendly. That's not a waterless solution, but it shouldn't use too much water if you're careful.

Get some of the extended-wear contacts for the duration of your trip. Use lots of eye drops.

Wow, you really don't have glasses? I can't imagine. I've been backpacking with contacts many times. I've never had much of an issue with getting my hands clean enough to take them out and put them in--my problem is doing it without a mirror, or with only a small signal mirror sitting on a rock or something. I always bring glasses for wearing after I stop for the day and take out my contacts.

It just sounds like you're a little too OCD about getting your hands really clean, or you're trying to carry all the water you need for your trip rather than refilling and purifying. If you bring CampSuds or some other similar soap you can get your hands clean enough in whatever water is around you (stream, lake). You might also try taking that waterless hand sanitizer. Just make sure you wipe the excess off and wait a while for your hands to dry. Use iodine or chlorine tablets to purify water from the wilderness, or get a filter like the Platypus CleanStream or something. Or just carry more water and drink less. For a two day, two night weekend backpacking trip four liters of water might be all you need if it's not super hot and humid, and if you can cook with whatever water is around you.

Really the same issue does not apply to bike touring. We're rarely limited in availability of water or ability to clean our hands. Maybe you should ask on a hiking forum.

I'm not just being OCD about clean hands, although I do have tendencies in that direction - in this case, touching my contacts with grubby hands has a very tangible effect on them and how well I can see.

I can go ahead and use some of my drinking water, but when you're in a middle eastern summer, drinking less is not an option and saving water for drinking is key when the next source of potable water is a couple hours' walk away! I can easily go through three and a half liters in a morning's walk and still run out and be thirsty before hitting the next water source. Anyway, I'll have to look into some of these water purification ideas to make sure that some of that natural springs I hit are safe to refill at, though I'd still be leery about drinking agricultural water.

wash your hands with environmentally friendly soap in questionably potable water, and then either rinse them off with a very small amount of drinking water or use some hand purifier gel. We're not talking about that much water, and if you need a little to maintain your vision, then you should simply treat the carrying of an extra water bottle as a necessary provision.

Also, when you say "middle eastern" summer, do you mean the middle east as in Saudi Arabia, or do you mean middle eastern US areas?

Or take along a pair of latex gloves and put them on before handling your contacts.

Middle east! Northern Israel to be exact. I never knew heat could be like this.

Hey thats a great idea actually.
Other than that, I would suggest going the glasses route. I got a really, really nasty eye infection once from dirty contacts. I would hate to think what would have happened if I was out in a remote area when that happened.
I know expense is an issue as the OP said, but its a lot cheaper than days in the hospital treating some crazy infection.

I use Accuvue 1-day Moist contact lenses. I wear them for a day then throw them away. And I always bring a couple of spares in case I need to replace them. The nice thing about disposable lenses is that you don't have to worry as much about contaminating them, since you only touch them when inserting and removing them.

As SBRDude suggests, you don't necessarily have to use "clean" water for the entire process of getting your hands clean. Here's what I'd suggest:

1) Wipe the majority of the dirt and gunk off your hands before you begin washing. You can use your shirt, pants, a rag, towel, or anything else you happen to have handy.

2) Use pre-packaged, cleaning cloths or baby wipes for the majority of cleaning. My favorite product is Dymon-brand Scrubs Hand Cleaners (not Hand Sanitizers). I buy them by the bucket and use them to clean my hands after working on cars/bikes/motorcycles. The Scrubs cloth is a bit rough and contains an embedded hand cleaner; one sheet is usually all you need to remove all of the dirt/grease from your hands.

3) Wash only the tips of the fingers that will touch your contacts with clean, potable water or saline solution. This is another advantage of disposable lenses: you don't have to scrub them around in your palm to clean them, so you really only need the tips of your fingers to be clean.