I've actually done quite a bit of research on this, and the reason is because as per the EOIR and the BIA, current interpretation of asylee/refugee law is that once a refugee/asylee adjusts to legal permanent resident, the definition of asylee/refugee no longer applies to the person AT ALL. Therefore you're no longer bound to the same benefits and restrictions that a refugee has. Once you become an LPR, you gain new privileges and responsibilities. One of those privileges is being able to travel without having to obtain advance permission from USCIS in the form of an RTD or AP. (This is the reason that RTDs are only valid for one year, they're meant to be temporary until you obtain a GC). As long as there wasn't fraud, the law doesn't distinguish between LPRs depending on how they obtained that status.
There is actually a case about this making its way through the Fifth Circuit Court of Appeals, but former court rulings seem to indicate this interpretation will stand.
The problem here is the "fraud" part. As many others have indicated, using or renewing your national passport could indicate re-availment. And even if the definition of asylee no longer applies to you as an LPR, this could make your life difficult during the citizenship interview if the officer "believes" there was fraud in the original application for asylum. This is where it gets sticky. The problem isn't using/renewing your passport as an LPR, the problem is this could arise a suspicion of fraud on the asylum claim, which gave you your green card. As long as your original claim of asylum wasn't based on persecution by the government, you "should" be fine. However under no circumstances should you go back to your home country until you become a US citizen. That is the biggest NO NO.
Know your case, know the law, consult with an experienced attorney on the merits of your individual case. And be ready to present a strong argument backed with evidence on why you KNOW using your NP isn't a problem as an LPR.
PS. I plan on using/renewing my passport after I get my green card.