If I change my tax rate, will it change the tax on old orders?

No. Each order saves its own tax when it is placed: the tax amount, each tax line with its name and rate, and whether prices included tax. Editing or deleting a rate on Settings → Store Settings → Manage Tax only affects checkouts from then on.

So you can safely:

1. Click Edit on the rate, enter the new Rate, and click Update rate.
2. Place a test order to confirm the new amount.

Old orders, and invoices printed from them, keep the tax they were charged.