If the pope renounces his US citizenship for the purpose of having diplomatic immunity or treats his acceptance of the papacy as an expatriating act with intent to relinquish citizenship within the meaning of INA §349(a)(4), he would not be inadmissible under the Reed Amendment: that amendment only applies when the reason for renouncing is to avoid taxation, and might not apply to relinquishment under §349(a)(4) regardless of reason since it uses the verb renounce rather than relinquish.
Why might the verb matter? The only parts of INA §349 that use the verb renounce are the ones about explicitly swearing or affirming an oath or affirmation of renunciation, not the other potentially expatriating acts. Relinquishment is the broader term in the statute which encompasses all such acts.
And I say "might" only because this amendment has been so rarely enforced that the courts haven't had occasion to rule on it. Only two people have ever been denied admission to the US under the Reed Amendment. It was a very badly drafted legal provision.
If the Pope were a mere diplomat, his immunities would be restricted to the acts directly related to his job in any country of which he’s a national or permanent resident. That’s because, unlike sovereign immunity, diplomatic immunity is based on a Vienna Convention full of restrictions like that.