Nope! In fact, it is against the law (the Sherman and Clayton Anti-Trust Acts, to be specific) to tie the purchase or use of a particular supply to the warranty.  In fact, if you want to see what HP has to say about it with your own eyes, check this out (but don't be surprised at the "marketing-speak" and scare-tactics they employ to try to convince you that you should only buy their over-priced, name-brand, brand-new cartridges):

Think about it:  Let's say your printer has a defective power supply.  How in the world could the manufacturer refuse to fulfill their obligation and fix their defective component (assuming the printer was still under warranty to begin with) just because you are buying a different brand of toner (or paper, paper trays, memory, etc.)?  They can't!

The manufacturer's ploy to convince you that you should only use their much more expensive cartridges is the same tactic the big automakers employ to try to get you to have your oil changed at the dealership instead of at your local "Quick-Oil-Change" shop (for 2 - 3 times more than the "Quick-Oil-Change" shop charges).

Why do those companies do that?  Do they really believe that the high-school dropout at the "Quick-Oil-Change" shop can't unscrew the drain plug in the oil pan as well as the high-school dropout employed at the dealership?  Of course that isn't their concern...  Their main concern is getting the repeat business and the huge profit to be made on the routine maintenance of your car... and the big printer, fax and photocopier manufacturers feel the same way about you and your toner cartridges!

As a case in point, we have over twenty HP laser printers in-house that have "eaten" a steady diet of our non-HP toner for their entire lives and not a single one has ever so much as "burped" funny as a result!  Millions of our cartridges have been used in thousands-upon-thousands of customer laser printers with the same results.  As we like to say, "The proof is in the printing!"