The 10NES chip can be effectively disabled by cutting pin 4 on the chip.  (The pins are numbered starting with the black dot as 1, and then continuing down the same side of the chip as the dot).

I did this to my old NES and it worked fine.  However, any hardware modification can be somewhat risky, so your mileage may vary.  

There's a picture of the chip and the pin to cut on [this wiki](http://console5.com/wiki/Improving_NES-001_Reliability) or there's also this [YouTube video](https://www.youtube.com/watch?v=LLMCj2VosLc) that goes step by step.

You can tell if the 10NES chip is disabled because the 10NES chip will reset the console over and over again if a valid cartridge is not inserted.  If your power light doesn't blink constantly anymore when the console is empty, then you did it right :)

As far as I'm aware, there's no downside to removing the 10NES functionality.  In fact, later NES models that were compatible with the same cartridges [did not include it.](http://en.wikipedia.org/wiki/Nintendo_Entertainment_System_(Model_NES-101))

For a more in-depth look at the 10NES chip, and a perhaps more in-depth way of disabling it (by routing pin 4 to ground instead), there's [this guide](http://nesdev.com/nlockout.txt).

However, some NES reliability issues are related to the funky 72-pin connector on the NES-001, rather than the 10NES chip directly.  You may also have to clean, refurbish, or replace this connector to improve the console reliability, depending on the particular issues your hardware has.