Your warranty with American Honda (USA) will NOT be voided if you do your own service/repairs to your Honda mower.
If, however, your service work causes some other problem or failure, that would not be covered under warranty. Example #1: You fix the recoil starter, and left wheel falls off. They are obviously not related, so a defective left wheel would be replaced under warranty. Example #2: You fix the recoil starter, and now the engine makes a scraping noise. Seem a nut off the starter fell off during your repair and got jammed under the flywheel. Fixing that problem would NOT be covered under warranty, as it was directly due to the something you did to the mower.
The Honda warranty covers factory defective parts only. Certain parts are defined as
consumable parts and when they wear, they are not considered defective. These include blades, air filter element, and recoil starter rope as examples. Each warranty job is a case-by-case deal; if your mower is only a month old and the starter rope breaks, it would probably be repaired under warranty. But, if you mower is 4-5 years old, and the starter rope is becoming frayed or weak, it will probably NOT be covered under warranty. You can read the full warranty document, including the full list of consumable parts here:
Honda Power Equipment Warranty
Recoil rope replacement is a simple repair; here's the procedure from the shop manual:
Finally, a brand new, entire recoil assembly, with new rope and grip, is only about $12.