No, your body knows how to produce NAD and won't stop doing it. However, as you get older, your body will need more NAD and/or produce less.
Some time back there was a rodent study in which they measured NAD levels, supplemented with an NAD precursor, and then measured NAD levels again post-supplementation. Because rodents age quickly, and NAD levels lower with age, you would expect that the post-supplementation NAD levels would be lower than the pre-supplementation NAD levels, but that would be due to age, not to supplementation, and that is exactly what they found. [I can't find the study, so if you know it, please send me a link!]
Nonetheless, people worry that once their bodies begin to rely on vitamins to produce NAD, they will stop producing their own NAD and essentially be hooked on the vitamin.
That's not how vitamins work, though.
We need vitamin C to avoid scurvy, and Vitamin A to avoid night-blindness, and Vitamin B3 to avoid Pellagra. We aren't addicted to those vitamins, and our body doesn't respond to their presence except by using or excreting them.
Human studies show that after supplementation ends, NAD levels drop to what they were before -- they are not further depressed:
Following oral administration of NMN or placebo, NAD+ levels were significantly increased in the NMN-treated group at 4-, 8-, 12 weeks, and then returned to basal levels after 16 weeks
Conclusion
Our bodies do not become increasingly dependent on any other vitamin when we supplement, and there is no reason to think it is true with NAD precursors. It is true that NAD levels drop over time, but that is not caused by supplementation, and indeed occurs regardless of whether you supplement.