Will this mess up my septic system, or leach lines?

Wholly ffuck!(excuse my French) but that is what a plumber would say out loud.  I hand wash dishes too.  . I fill the sink over half full of soapy water and wash the dishes .  As it is a double sink  I rinse on the other half and into the drying rack.  If your fingers squeak when you rub on a wet dish = it is clean.  That is dishes, pots pans, cups, bowls, mugs, forks, spoons, knives. Soapy one sink and rinse in other sink.

. Laundry is done once a month or so.  Showers are 10 minutes and I am done.

Your septic tank is probably 1000 gallon(the small ones are 600 gallons.    The tank holds the solids from your poop chute, and kitchen food and your hair and dead skin cells and scabs and TP.   If you use a garburator then that food also goes into the septic tank as solid materials, and dirt and soil from gardening off your clothes and bits of clothes and that is about it.

.   I don't have a garburetor, nor do I use the toilet as a waste basket.  Nor am I heavy with bleach. I rarely use it.  Nor do other chemicals go into the system.

.  I am only asking this because I was talking to plumbers who say they NEVER clean their tank. It works just fine.

A leech field aka septic field aka drainage field if you are in decent ground (meaning not clay and the bottom of a valley or ravine) will absorb the water coming out of the sink, toilet, bathtub, shower, laundry tub and that water will go into the soil lower than your basement floor(unless you are in a flood plain and this year the water is up and is seeping into the basement(meaning it can't go down when it is flooding up.) Then YOU have problems.   Most homes are built on ground that is not swamp land (so not the lowest point)  When you are above that then water will always go down...as it does everytime it rains.

. TP does break down, but slowly. Paper such as newsprint won't break down at all. Neither will pampers or Maxipads or Pantiliners.  So those things remain solids in the septic tank, filling it.  Basically what comes out your lower backside is what goes in the toilet.  That is it.

10 years go by and I see it is not even half full. I figure for 1 person it would take 40 years before it needed pumping out.   It is Full of water, yes but that just goes into the drainage field to drain away to HELL or Lower.(like China) You got the concept.

. You will NEVER FILL IT.  Your way definitely 100 years can go by before it needs pumping out.

I've been washing dishes that way for almost 30 years, and both of the septic systems i had in that time have been fine.

your septic system will be just fine with this -- grampa

90% of leach field flooding and septic tank issues are caused by doing laundry.

This is why most people dig a separate pit and run the washing machine drain water to it instead.

Over-consumption of water can cause problems, but I doubt the amount of water you're using to wash dishes is going to push you over the threshold.  

Of course we do not know your septic system's capacity, how many people you have living there and how much water residents are using for other things like showering, how efficient your appliances like washing machine is and whether you're running a bunch of things at the same time...

There are far worse things you could be doing to your septic system than hand-washing dishes.

Invest in a dishwasher. No hassle and gleaming dishes. They use less water than hand washing too.

Your septic probably contains thousands of gallons - a few more won't hurt it.  DON'T run fibrous things down the disposer that the system can't "digest" such as coffee grounds, bones, too many stringy veggies like asparagus  (ugh) orange rinds and such. . . . use septic friendly laundry detergents, minimal bleach, no paints, paint thinners, oil, gas, etc.  Consider occasional use of a product such as Rid-X to reinforce enzymes and bacteria in the system.

I recently sold a house . . . lived there 23 years . . . the previous owner had clogged the leach field by running coffee grounds down the drain for years . . . . . entire system had to be replaced.  When I sold the buyers had the tank pumped and inspected . . . there was less than an inch of sludge at the bottom after 23 years !

Are you usng a strainer in your drain so debris don't go down?

The more water one puts in the system, the more work that leaching field has to do. Some work fine forever while others have issues after just a few years.